At very, very low levels, there are some options for climbing. The best of them are three related abilities:
Spiderwalk, a least invocation for warlocks (Complete Arcane),
Spider climb, a 2nd-level spell for druids, sorcerers, and wizards,
Dance of the spider, a 3rd-level Shadow Hand stance (Tome of Battle, natively available to swordsages from the same).
Spiderwalk and dance of the spider are continuous effects, while spider climb lasts 10 minutes per level. Since spiderwalk is both permanent and available from 1st, it is easily the best option. By the time swordsages get dance of the spider (5th level minimum), climbing as a whole has been largely obviated by flight.
You could also consider getting spider climb as a racial ability, though the options there are very limited, and extremely expensive. Jungle goblins from Unearthed Arcana and grippli from Dragon Compendium can instead get a Climb speed, which isn’t as good as spider climb, but still far better than nothing, and those are LA +0 races so they aren’t crippling like the spider climb races are. A single level of ape totem barbarian can also get it, but that comes at a high cost—not just the level itself, but the ape totem is incompatible with the excellent lion spiritual totem from Complete Champion—which is extremely important to many melee characters because it offers pounce, i.e. the ability to move and full-attack in the same turn.
Anyway, all of the above requires at least one hand dedicated to climbing. Barring something like the insectile template (Dragon Compendium), that means you only have one hand left for attacking. Meanwhile, almost every mundane option for substantial damage output requires both hands. Even a hand crossbow—which can be fired with one hand—requires another hand for reloading. And benefits immensely from dual-wielding. You could try some other “hands-free” weapons, like unarmed strikes, spiked armor, or the hidden weapons from Complete Scoundrel, but those are all awkward weapons to use. So no matter what you do, climbing is a huge drawback.
Notably, warlocks get the eldritch blast invocation at 1st, a basic ray attack that’s pretty similar to archery, which combined with spiderwalk gets you pretty close to the character you want in one turn, and unlike true archery, eldritch blast does not care how many hands you have free. Combining eldritch blast with sneak attack is awkward, though unseen seer from Complete Mage and/or arcane trickster could at least allow you to progress both. If other people have to climb and don’t have a Climb speed, that could even work since they’ll lose their Dexterity bonus to AC (thus allowing you to trigger sneak attack). The eldritch glaive invocation could potentially also allow it to be used in a more melee-oriented situation (and also allows for iteratives, so you get more of your sneak attack).
Also worth mentioning is the Up the Walls psionic feat, which allows you to, well, go up walls, as long as you end your movement on some flat surface. Arguably, that gives you an opportunity to use both hands on your weapon while you move—maybe. You could at least combine it with Spring Attack, which you could try to optimize using the Bounding Assault and Rapid Blitz feats from Player’s Handbook II. That is a very weak, very expensive strategy, and utterly reliant on there being regular ledges for you to use. But maybe your DM will allow a combination of Up the Walls and spider climb to be used to allow you to briefly remove your hand from the wall to attack normally, obviating the need for ledges or wasting feats on Spring Attack et al. Then adding a level of warlock and the Up the Walls feat to basically any other build could work. This is dependent on your DM, though, since the rules offer nothing specific about that combination.
My suggestion ends up being something like this:\$\newcommand{\d}{\mathrm{d}}\$
\begin{array}{l|l|l|c|c|l}
& & & \textbf{Sneak} & \textit{Eldritch} \\
\textbf{Level} & \textbf{Class} & \textbf{Special} & \textbf{Attack} & \textit{Blast} & \textbf{Feat} \\ \hline
1^\text{st} & \text{Rogue} & & +1\d6+1 & - & \text{Craven}^1 \\ \hline
2^\text{nd} & \text{Warlock} & \textit{Spiderwalk} & +1\d6+2 & 1\d6 \\ \hline
3^\text{rd} & \text{Trickster}^2 & \text{Steal spell (0 or 1st)} & +2\d6+3 & & \text{Weapon} \\
& \quad\text{Spellthief} & & & & \quad\text{Finesse}^3 \\
4^\text{th} & & \textit{Detect magic,} & +2\d6+4 \\
& & \quad\text{steal spell effect,} \\
& & \quad\text{1st-level spells} \\ \hline
5^\text{th} & \text{Rogue} & \text{Penetrating strike}^4 & +2\d6+5 \\ \hline
6^\text{th} & \text{Unseen Seer}^5 & \textit{Eldritch glaive}^6 & +3\d6+6 & & \text{Travel Devotion}^7 \\
7^\text{th} & & \text{Advanced learning}^8 & +3\d6+7 & 2\d6 \\
8^\text{th} & & \text{Least invocation,} & +3\d6+8 \\
& & \quad\text{div. spell power +1}^9 \\
9^\text{th} & & & +4\d6+9 & 3\d6 & \text{Martial Study}^{10} \\
10^\text{th} & & \text{Advanced learning,}^8 & +4\d6+10 \\
& & \quad\text{guarded mind,} \\
& & \quad\text{lesser invocation} \\
11^\text{th} & & \text{Div. spell power +2}^9 & +4\d6+11 & 4\d6 \\
12^\text{th} & & \text{Lesser invocation} & +7\d6+12 & & \text{Martial Stance}^{11} \\
13^\text{th} & & \text{Advanced learning}^8 & +7\d6+13 & 5\d6 \\
14^\text{th} & & \text{Div. spell power +3,}^9 & +7\d6+14 \\
& & \quad\text{Lesser invocation} \\
15^\text{th} & & \text{Greater invocation} & +8\d6+15 & 6\d6 & \text{Master Spellthief}^{12} \\ \hline
16^\text{th} & ???^{13} & & +8\d6+16 \\
\end{array}
Race doesn’t really matter, though whisper gnomes from Races of Stone are ideal for stealth and work well here. A kobold playing to their strengths is also a very good race here.
Craven is a feat from Champions of Ruin that adds your character level to your damage on any sneak attack. Very useful. Cannot be used if you have or gain immunity to fear, however, which becomes painful at higher levels—consider retraining. Also make sure you ask your DM if you can ignore aura of courage if there’s a paladin in the party.
Trickster is a spellthief substitution from Dragon vol. 353, pg. 85-86, that trades 2 skill points per level, trapfinding, and all sneak attack improvements after 1st level for a spellcasting progression like a bard as well as the addition of all bard spells to the spell list. Mostly, it’s 1d6 sneak attack plus 1st-level spells to qualify for unseen seer, and Master Spellthief is kind of useful later. Could be replaced with just another level of rogue plus a level of anything that gets 1st-level arcane spells at 1st level, like sorcerer, wizard, or duskblade (Player’s Handbook II).
You have no particular use for Weapon Finesse at this level, but you’ll want it later for eldritch glaive and/or Eldritch Claws.
Penetrating strike is an ACF from Dungeonscape that allows you to deal half your sneak attack damage to things normally immune to sneak attack, replacing trap sense. Trap sense is garbage, while this allows you to maintain damage output by things that usually screw you over. Clearly worth it.
Unseen seer from Complete Mage requires 1st-level arcane spells, hence the dip in trickster spellthief. We don’t have to pick spellthief for the spellcasting advancement, however—Complete Arcane allows warlocks to benefit from prestige classes that advance arcane spellcasting classes. So unseen seer here is advancing warlock. It also gives ¾ BAB, which is nice for us since we want interatives.
Eldritch glaive is a least invocation from Dragon Magic, that allows you to make melee touch attacks with 10-ft reach that deal your eldritch blast damage. Unlike hideous blow, which is a standard action for a single non-touch attack, eldritch glaive is a full-round action and you get to make iterative attacks and make attacks of opportunity with it. That means more eldritch blast damage, more sneak attack damage, etc. etc. This invocation is basically what makes this build work past approximately 8th level (when you first get iteratives).
- Note: it may be possible to replace eldritch glaive with the Eldritch Claws feat from Dragon vol. 358, especially if you go with a kobold and take Rapidstrike and Improved Rapidstrike from Draconomicon. Using unarmed swordsage and the Beast Strike feat from Dragon vol. 355 instead could be even better.
Travel Devotion is a feat from Complete Champion that allows you to, once per day, move as a swift action for the next minute. That allows you to move and attack, which is really important (and it’s compatible with eldritch glaive, unlike pounce). If you had turn undead, you could use this more—and clerics can get it as a bonus feat, which makes it strongly worth considering a dip in cleric instead of the 3rd level of rogue.
Advanced learning can be used to learn any divination spell from any spell list, as long as you can cast it. You can only cast 1st-level spells, since we aren’t advancing spellthief, but that’s good enough to learn the excellent sniper’s shot, a 1st-level ranger spell from Spell Compendium that eliminates the 30-ft. restriction on sneak attack. The 2nd-level *hunter’s eye& from Player’s Handbook II would be even better, but requires advancing spellthief some.
Divination spell power increases your caster level for arcane divination spells by the listed number, and also decreases your caster level for non-divination arcane spells by the same amount. But since almost-everything you do is an invocation, not a spell, you don’t care. It does boost your caster level with hunter’s eye, though, so that’s nice.
Martial Study from Tome of Battle allows you to learn one of the martial maneuvers from that book and use it once per encounter (once per minute outside of combat). One of them is cloak of deception, a 2nd-level boost that turns you invisible (per greater invisibility) for your turn. Great way to trigger sneak attack from a distance. Also important because it qualifies you for...
Martial Stance, also from Tome of Battle, which allows you to learn one of the stances in the book as long as we already know a maneuver from that discipline. In particular, we are interested in assassin’s stance, a 3rd-level Shadow Hand stance, which grants +2d6 sneak attack damage. Luckily, cloak of deception is also from Shadow Hand, so that works out nicely.
Master Spellthief from Complete Scoundrel lets all your arcane class levels to stack for the caster level of all those classes. So with this feat at this point, both spellthief and warlock have caster level 13th.
Options for 16th and beyond are basically anything that advances sneak attack or anything that advances spellcasting. Abjurant champion from Complete Mage would be a great choice, for example, boosting your BAB along with invocations. Perhaps even better, progress spellthief another couple of levels to get 2nd-level spells (probably earlier, with a couple of the unseen seer levels, so that you can take hunter’s eye with one of the advanced learning opportunities). Then you can progress both eldritch blast and sneak attack with arcane trickster. Getting 2nd-level spells would also open up eldritch theurge from Complete Mage, to advance warlock alongside spellthief spellcasting, if you wanted to do that—with Master Spellthief, that quickly leads to silly caster levels, which is good for hunter’s eye.
Anyway, the long and short of it is that you can get climbing to work, as long as you’re using a magical attack, but it’s expensive and doesn’t quite seem to match what you were looking for. Which brings us to another important fact: ultimately, none of the above is worth paying much attention to at all, because your campaign is going to go as high as 16th-18th level. A 16th-18th level character should never be climbing. Characters can gain the ability to fly as early as 5th, and flight is all-but-mandatory by around 10th level. Flight never gives you any trouble using your combat options, nor does it require a free hand.
Dragonborn (Races of the Dragon) and raptorans (Races of the Wild) can fly as a racial feature, at 6th and 5th, respectively. Many spellcasters, of course, gain fly and then overland flight. Warlocks (Complete Arcane) generally find the fell flight invocation mandatory at 6th—in fact, the build above has absolutely no reason not to take fell flight at 10th, and many, many reasons to do so (ask the DM to retrain spiderwalk at that point). Characters who get flight from neither race nor class have to buy it, but that is still generally a thing they must do. The feathered wings graft from Fiend Folio is the most cost-effective option, but those drive you mad unless you’re evil, so that’s a concern. A winged mask from Magic Item Compendium is probably the cheapest general-purpose version.
In short: climbing is very, very bad for you, and there are limited options for making it better. Flight is too accessible, and too otherwise-mandatory, to bother with even those options that do exist—after all, the best one is spiderwalk, which comes from the same class that gets fell flight five levels later. And since every character needs to be able to fly by mid-to-high levels, it’s not really something to optimize around so much as just a box for every build to check. Too many options to seriously suggest any one build that flies.