Instructions
Focus exclusively on using only your bow in combat. Using melee weapons or offensive magic will only increase the amount of time to level up your "Marksman" skill.
Equip a weak bow for your level. This will give you the opportunity to fire the bow more often when fighting monsters, increasing your chance of obtaining a skill increase.
Use poisons or magic that drain strength or add encumbrance to your enemy. This will slow your enemy, allowing you to get more shots in.
Avoid sneak attacking weaker enemies with your bow. A sneak attack will often kill a target outright, leaving you little chance to level your marksman skill.