As unfortunate as it is, chances are the only way you can get maximum output using a melee-based Warlock without being accused of being "cheese" is if you do indeed have access to the Hexblade Patron. That being said, I wrote out the optimal way to deal damage without being gimmicky. For each of these I used point buy to properly spread out stats. However, it is possible to do it using just the standard array.
For all the damage estimates below, I'm going to assume the AC of the target is 15. This is about average for those with heavy armor, and below average for those with a shield.
Race: Human (Variant)
Class: Warlock (Any patron, Pact of the Blade)
Point Buy: 15, 15, 14, 10, 8, 8
After Bonuses: 16 (Str), 16 (Cha), 14 (Con), 10 (Wis), 8 (Int), 8 (Dex)
Feat: Polearm Master
Notes: Going for Strength and Polearm Master is better than going for a Greataxe for instance because it means you get to proc hex twice. Once for the main weapon, and once for the hilt. This even scales nicely since when you get Thirsting Blade as an Eldritch Invocation at level 5, you can still use your Bonus Action for the bash with your hilt. That, and reach weapons are great for casters who use melee weapons since they can still cast spells freely on the target when they aren't in traditional melee range.
Turn 2 Onward (Same Target):
Action: Glaive (1d10 Slashing + 1d6 Necrotic)
Bonus Action: Polearm Master Bash (1d4 Bludgeoning + 1d6 Necrotic)
Average Damage: 7.95 ((40% Hit × (9.00 Average Roll + 3 Str)) + (5% Crit × 9.00 Average)) + ((40% Hit × 6.00 Average Roll) + (5% Crit × 6.00 Average))
I actually think this is a viable way to play Warlocks and isn't even considered cheese. Humans make pretty fun Warlocks, and being strength based doesn't even cripple your ability to play. You can even make it so you don't dump your other three stats and reduce your constitution a bit to compensate the other stats. For that you point buy you would have 15, 15, 11, 11, 10, 9.
If you want to go more traditional and aren't worried about not getting your feat until level 4, you could go for a Half-Elf and for stats go 15 strength, 14 charisma (which turn into 16 strength, 16 charisma), put your remaining point buy and 1 racial point wherever you want, and then grab Polearm Master at level 4. However, going with Human (Variant) means you can actually +2 your Strength or Charisma at level 4 and still have the feat.
PHB + 1 (XGtE)
Race: Human (Variant)
Class: Warlock (Hexblade Patron, Pact of the Blade)
Point Buy: 15, 15, 13, 11, 9, 8
After Bonuses: 16 (Dex/Str), 16 (Cha), 14 (Con), 12 (Wis), 10 (Int), 9 (Str/Dex)
Feats: Dual Wielder
Invocations: Improved Pact Weapon, Fiendish Vigor (if Strength) or Armor of Shadows (if Dexterity). At level 5, pick up Thirsting Blade. Also at level 5 if you want more damage rather than survivability, get rid of Fiendish Vigor/Armor of Shadows and substitute in Eldritch Smite (for single target) or Maddening Hex (for AOE).
Notes: This is definitely the build that is the least gimmicky and allows you the most options for what to do. It is what I play the most when I'm not a DM and I've spent quite a bit of time tweaking it to find a good balance of things.
Additionally, you might notice that I just have written in "weapon" most places, and that's because this build works for being Strength or Dexterity based. Personally I very much enjoy being Dexterity based, but Strength gives you a lot more flexibility with what weapons you can use, and allows you to do fun things with Grapple, Shove, and the like.
You can use Hex Warrior on a specific weapon in addition to those that are summoned from Pact of the Blade. This means that after you finish a long rest, Hex Warrior can be used on a weapon you use in your off-hand, and then have the pact weapon in your main-hand. As such, you can easily dual-wield and still use your Charisma modifier.
Action: Pact Weapon Attack (1d8 Slashing/Bludeoning/Piercing + 1d6 Necrotic)
Bonus Action: Other Weapon (1d8 Slashing/Bludeoning/Piercing + 1d6 Necrotic)
Average Damage: 9.4 ((45% Hit × (8 Average Roll + 3 Str/Dex + 1 Improved Pact Weapon)) + (5% Crit × 8 Average Roll)) + ((40% Hit × 8 Average Roll) + (5% Crit × 8 Average Roll))
This is a great build with high Charisma (obviously) and allows the flexibility of being either Dexterity or Strength based, depending on your character and your preference. It scales nicely because of what you pick up at level 4 and the fact that you are able to put your ASI into stats rather than a Feat. Also, it opens the window for you to be very dynamic in your play-style since you are still able to fall back on Polearm Master instead of Dual Wielder if you prefer to be at range.
Level 9 Example Build
Starting at Level 3:
Race: Human (Variant)
Feat: Dual Wielder
Non-Combat Character Design:
You must already have a character concept in mind, otherwise you wouldn't have posted such a specific question. However, I just want to put in my two cents.
As for why a character might want to swap between weapons, I have a few ideas that I actually might borrow myself later on since they all seem like a blast. They all have a twist on what Charisma is, since I have always disliked the idea the Charisma encompasses everything, and that a high Charisma character always is outgoing, boisterous, etc.
- Maybe they spent their whole life training in different types of combat before they become a Warlock so they enjoy showing off what they can do. This character would be Charismatic through their pompousness and could end up getting on the nerves of other characters, but would be hilarious if done right to the other players themselves.
- Maybe they were a wannabe Lore Bard who lived the idea of being a Jack of All Trades. When that didn't work out, they became depressed and started researching ancient magic and stumbled across a Hexblade and then made a pact.
- Maybe they were just some average person who, no matter how many adventures they go on, is still extremely entertained by the fact that they can summon weapons out of thin air so they never want to settle for just one. If you go with this concept you could even have them grumble in the morning when they have to mark their non-pact weapon with Hex Warrior saying things like, "why do I always have to pick just one for the day."
As for how/why you might do this in combat, different weapons become useful through an encounter. Rangers routinely switch from ranged to melee and back again. The only downside is that you would have to give up your turn. This isn't necessarily a terrible thing. Say for instant you want to confuse your enemy or have a bit of fun. If the opponent has already used their reaction, you could toss your weapon at them (not an attack, just dropping it), run away from them, and then summon a Longbow and start shooting on your next turn.
This is the part that is potentially the most important. DMs already have a lot to keep track of; it's not an easy role. To make your DM's life easier, you could always use a weapon with the same damage die, only switch off at the end/start of a session, or create macros to handle the math for each type of weapon beforehand if you are using something like Roll20. The first option there is probably the best, but it isn't as limiting as you'd think. As it is, most one-handed weapons already use 1d8 for their damage unless it's a light weapon such as Daggers and Shortswords.
Hexblade offers some of the most flexibility of any class besides Mystic from UA and some multi-class abilities. You can blast people from range or get up in their face with Dual Wielded weapons and put the hurt on. Additionally, if you pick up a +X magic weapon, you can use your pact weapon in your off-hand and your new magic weapon in your main hand which allows you to reap the benefits of Improved Pact Weapon as well as the magic item.
Hope this helps you in the process of making your character! I'd love to hear how it turns out.