My thoughts go like this.
Class: Slayer. Gains Some useful things like sneak attack, favored target, and some terrain bonuses as well as Full BAB.
Race: I'm a bit biased towards humans (for the bonus feat), but any standard archer type should work nicely (elf, human, the small races, anything you'd want that nets useful bonuses to archers)
Level 1 feats: mounted combat, weapon finesse
Bow: I'm pretty sure you can only use a shortbow whilst mounted, so there's your main weapon. If there's a way to use a longbow mounted without major penalties then do it.
Armor: any light armor that won't mess up your ride skills a lot.
Secondary Weapons: Mostly for when/if you end up on the ground in melee
If you pick Elf: Elven Curve Blade (a d10 2 handed weapon with finesse and an 18-20 crit range). if the DM will allow 3.5 material, get an elven thinblade, elven lightblade, or both, they're good 1 handed finesse weapons with 18-20 crit range.
If not: stick with rapier and a light weapon
Either way: Lance. Jeepthingjim above me is right that charging 3x damage is worth your time. Though you better have high strength.
Slayer talents: This is the main draw for this since your higher speed on any decent speedy mount will let you flank easier.
Assassinate: if you study your target for one round (and they're a favored target) Your sneak attack has a chance of killing them instantly. Though this is an advanced trick, so you won't get it for a while.
Terrain Mastery and Slayer Camoflage:
Terrain Mastery (Ex): A slayer with this talent chooses a favored terrain on the ranger’s Favored Terrains table (Pathfinder RPG Core Rulebook 65). The slayer gains a +2 bonus on initiative checks and Knowledge (geography), Perception, Stealth, and Survival skill checks when he is in this terrain. A slayer traveling through his favored terrain normally leaves no trail and cannot be tracked (though he may leave a trail if he so chooses). A slayer can take this ability multiple times, each time applying it to a new terrain, and increase by +2 the bonuses granted with all favored terrains previously taken. These bonuses don’t stack if the slayer is in terrain that falls into more than one category of favored terrain.
Slayer Camoflage:(Ex) A slayer can use the Stealth skill to hide in any of his favored terrains, even if the terrain doesn’t grant cover or concealment. A slayer must have the terrain mastery talent to select this talent.
So with those talents, there's your stealth request.
Slowing strike: halves an enemy's movement speed for a d4 rounds
Hunter's Surprise: Lets you pick one enemy to add your sneak attack damage to the damage for every attack you make against them for the round.
Optional slayer powers
fast stealth: I'm not sure if this affects a mount. So I put it as optional
Firearms training: Since this doesn't count as archery you may ignore this, but it may be worth it if you can use it. Your call.
Grit: related to the firearms training. Using grit may be a good little bonus to your guy
Feats after first level in no particular order:
and other mounted feats you need
your standard combat feats: (weapon focus, improved critical etc.)
Improved critical (secondary weapon) critical focus, and critical feats. I know it seems strange to say spend feats on secondary weapons (or at least make them keen). but your secondary weapons (assuming you chose the elven stuff or the rapier) can get a crit 15-20, meaning you'll be able to use critical feats a lot using them.
+6d6 sneak attacks you can get via flanking
killing enemies with master slayer by 20th
Special powers like assassinate can only work on favored targets
stealth without cover requires specific terrain.
Relatively feat intensive from not using fighter