How can I improve Light Armor as an Archer?

There are three trainers for light armor (source):

• Scouts-Many-Marshes | Located: Windhelm | Adept (0 - 50)
• Grelka | Location: Riften | Expert (0 - 75)
• Nazir | Location: Dark Brotherhood Sanctuary* | Master (0 - 90)

**Nazir eventually moves to the Dawnstar Sanctuary if you are doing the Dark Brotherhood quest*

@Layke outlines an effective way to use a trainer to level up without needing to worry about money, and also level up your pickpocketing skill at the same time. You just train, then pick pocket back your money, rinse and repeat.

A method to level light armor without making use of a skill trainer is to find a weak enemy (like a mud crab), and just let them attack you for awhile. If you use a healing spell to keep your health up, you shouldn't run into any problems. Using this method you'll be able to level up your Restoration skill as well.

A really easy and fairly safe way is to buy a house, stand in front of it, and shout at the guards.

They will fight back and try to kill you. Just stand there and take the abuse, and when your health gets low walk into your house to let it refill. Walk back out to the waiting mob of guards and repeat the process.

It will take a while and you may develop a bit of a fine, but if you can afford it it's worth it, especially compared to how much the trainers cost.

Getting hits means getting skills point.

You want to take damage, as fast as you can survive. For such low skill levels (below 40), I recommend getting shot at by bandit archers.

Get your best light armour, turn difficulty settings to novice, find a reasonably easy enemy and let them attack you. Use your healing spell, which is available from the beginning of the game, if your life gets too low. Doing this will get your restoration up as well as light armour. Do not use a shield as it will contribute to block instead.