Horse Archery is about as much fun I know how to have legally. Horse Archers work well on large maps (ie don't fight in forests or mountains) against light to medium armored foes. They're fragile (poor to none head armor) and sometimes charge smack into large blobs with pointy sticks and die. However, they're both cheap and fast as hell on the world map so you never have to fight if you don't want to. You can read about skirmish cavalry in general on the Return of the Sarranid Cavalry thread,
http://forums.taleworlds.com/index.php/topic,132068.0.html. Basically, you should try to disperse the enemy before ordering a charge but when fighting Bandits of any kind (not Sea Raiders!) or archers just send the boys in and watch the chaos. Khergit Horse Archers will not win against lots of heavy cavalry, you need to have a lot of Lancers for heavy foes.
As for playing a horse archer I would suggest using this in character creation: Father Nomad, Street Urchin, Poacher, Wanderlust. Put all proficiency points into Archery until you reach 150. You want a high Str (for Powerdraw) and high Agi for Riding and Horse Archery. Get PD 3, Khergit or Bodkin arrows and a Khergit Bow asap. The reason I don't like Strong Bows or Warbows is their slow speed which absolutely matter when you ride. You can also upgrade to Strong and Masterwork Nomad or Khergit bows as you level up PD. You can equip two quivers and use a twohander or Hafted Blade as your melee weapon so you can shoot until everyone is dead. Horse archer battles take time.
Preferably you want to have a higher Ride skill than your mount needs as this affects the size of the reticule and how well you handle the horse. Some like the Courser for it's speed, I prefer Desert, Steppe and Sarranid horses for their maneuver. Of course, the Steppe Charger is a great archer platform when you have riding at 5 or above but well, it's not fast enough for my taste. The faster you ride, the larger your hit reticule will become. Slow to a trot and shoot following infantry in the face as they pursue you. This is how you gain proficiency points in the early game.
My aim when playing a HA is to never ever get hit which means moving at top speed all the time and changing direction often. There is, sadly, no penalty in Warband for using archery wearing full plate, a Vaegir War Mask and steel gauntlets. I play with open helmets or coifs, leather gloves, light mail boots and never wear armor that weigh more than 20. Just because I want to.
For firing you will need to learn the art of deflection shooting, basically it's a lot like air combat without the barrel rolls. So you need to put the arrow where the target will be when the arrow has reached it plus account for your own speed and direction. Practise. Get a high bow skill. Attend tournaments in all Khergit towns. Try to take the hard shots. If you have a higher PD than the bow requires your reticule will be stable longer.
If you have more specific questions, fire away.
