What is the best way of leveling restoration?

Skills only level when the game judges them to have been "usefully" employed; in the case of spells, that means the spell must have had some noticeable effect. Casting healing spells at full health does nothing, nor does casting damage spells into a wall or armor spells without entering combat.

For leveling restoration, your best bet is to use the turn undead spell line, which gives skill increases very quickly. Head to your favorite barrow and start sending the skeletons fleeing before you kill them.

I would strongly caution against "boosting" non-damage skills in Skyrim. You risk your character leveling up too rapidly without having the combat skills he needs to survive. These techniques are best employed to catch up support skills that have fallen behind.

There are actually many techniques to leveling restoration in Skyrim. I found myself needing to do this in order to balance my character better; I had completed the main quest line so was merely playing for the hell of it.

How it works

The trick for leveling spell-based skills is to cast the spell so that it will have some affect or rather: change the environment/game.

In the case of restoration spells and therefore leveling it, you'll need to cast a restoration spell -- Healing, Fast Healing, Close Wounds for instance -- so that it gives you health. Casting on a full health bar won't give you experience.

To summarise: the general idea is to take some damage and then cast a restoration spell.

Leveling restoration by spells alone

So, one way in which to harm yourself so you can heal straight away; and do so in a controlled manner, is to cast the alteration spell Equilibrium -- which converts health to magicka for the duration of the cast, but can kill you -- and then heal once your health is low enough for the restoration spell to apply.

I personally found this a little slow so opted for the slightly modified method below.

Taking constant damage

An alternative method is to stand on a device or area that will harm you constantly, such as a under a fire trap or as in this example the wind at the top of High Hrothgar that leads to the Throat of the World. In the latter case (slight-spoiler alert!) you just have to walk towards the winds to stumble and take damage, at which point you can dual-wield a restoration spell and cast as soon as your health gets too low.

You could also find a pack of wolves, a bear (if you can take the damage), spiders or other pack-based enemies and not attack but dual-wield your restoration spell so you don't die. This saves on having to cast two spells and has the added benefit of leveling armor and such skills. Do bear in mind though, that leveling multiple skills like this will yield your overall level increasing which could mean you'll end up fighting mobs that are too high for your combat skills.

As I said above, I opted for the winds at High Hrotgar as you only need to concentrate on casting the restoration spell and the damage is constant which makes it all a little faster to do, unlike the Equilibrium route.

You could always just use flames and healing at the same time on shadowmere (poor shadowmere he's just a big ol' punching bag). Hope this helps =)