Work With Trainers
Five trainers across Cyrodill will, for a price, raise your marksman skill. The two low-level trainers are Elda Dark-Heart and Shameer. Elda lives in Bruma in the house directly behind the Bruma Chapel. Shameer's house is in Skingrad across the from the front entrance of the Skingrad Chapel. The midlevel trainers are Reman Broder and Pinarus Inventius. Reman's house is right behind the Two Sisters Lodge in Skingrad; Pinarus' house is the second house on your left as you enter Anvil from the main dock gate. The master trainer is Alawen. Unlike the four other trainers, she requires more than money before she will agree to train you. You will need to bring an elven bow to her at Troll Candle Camp just outside of Anvil if you want her training.
Wear Enchanted Items
As you loot treasure or shop around in the various stores, keep you eyes open for clothing, armor or jewelry with enchantments to boost your marksman skill. If you're not finding anything like this in your treasure raids, or you don't want to pay the expensive prices in the stores, you can always enchant your own items. You will need access to the Arcane University's enchanting alters and some filled soul gems. Once you have your wearable items enchanted with marksman enchantments, all you have to do it wear them to get the boost.
Hit the Books
There are hundreds of books waiting to be read all across Cyrodill. Certain books will increase a skill when you read them. Five books will increase your marksman skill. Two of these books, "Gold Ribbon of Merit" and "The Black Arrow 2," are located in the Dark Brotherhood Sanctuary in Cheydinhal. "A Dance in Fire 5" is at the home of the low-level trainer, Elda Dark-Heart. "Father of the Niben" is in Castle Bravil, and you can find "Vernaccus and Bourier" at Renoit's Books in Chorrol.
Controlled Field Practice
Sometimes you are only lucky enough to get more than one shot in before your opponent is too close for further arrow attacks. Try to get out of his reach completely. Look for ledges in caves that will offer a safe perch so you can freely fire on you enemies as they struggle to reach you. Or wait behind a closed dungeon gate that has space enough for your arrows to fly through. Some Goblin caves have pens full of rats that make excellent target practice while increasing your skill level at the same time.