What's the best way of doing this? I'm currently level 6 but have already encountered some adept and expert level chests. Hoe an I go about quick leveling lockpick to make these doable without using 10 picks.

Also, where can I buy additional lockpicks?

Thanks.

thieves guild has lockpicks. pay for training if you must. thieves guild also has 'practice' chests down there but i'm not sure they count towards skill leveling. just work on it a bit. once you get it figured out it becomes much much easier.

Lockpicking in this game is fantastic. You do level up your lockpick skills a little each time you attempt to pick a lock, even if not successfully.
I've never bothered picking perks for lockpick but I've found it's just so high on it's own I've never found it nescessary.

Best way to level up lockpick skills is to pick the lock on like, everything. Dungeons are awesome for this. Locked empty cells. Locked doors wirh a few miserable potions inside.
It really is about using the skill every opportunity you get, even on mundane things.

Lockpicks are alot easier to find than the wretched bobby pins of Fallout, which is good news!

Often every bandit you kill will carry one or two. You'll also find them in chests, and keep your eyes peeled as they can be found in barrows and dungeons just lying around, allthough they're pretty small and hard to see.

You can also buy them in small quantities from traders. Warmaidens allways has some for sale in their misc section, and the trader in whiterun and riverwood sell them too.
I'm assuming most common goods traders will sell them in their misc sections.