Lockpicking is the ability with which you can open locked chest or gates / doors. 
This simple guide offers you different locations all the way from 1 to 350 depending on your current level with a few different options:

Lockpicking 1 to 70

HORDE
Skill range: 1 to 70 
Location: The Barrens 
Item to pick: Buccaneer’s Strongbox (Buccaneer’s Strongbox)

HORDE
Skill range: 1 to 70 
Location: Ghostland 
Item to pick: Burial Chest (Burial Chest)

ALLIANCE
Skill range: 1 to 70 
Location: Redridge Mountains 
Item to pick: Practice Lockbox (Practice Lockbox)

Lockpicking 71 to 170

HORDE
Skill range: 71 to 170 
Location: The Barrens 
Item to pick: Gallywix’s Lockbox (Gallywix’s Lockbox)

ALLIANCE
Skill range: 71 to 170 
Location: Westfall 
Item to pick: Duskwood Chest (Duskwood Chest)

TIP: One chest is located on the top of the tower, left of the table found there. Unfortunately, there is an elite close by, therefore bringing a friend is advised. Respawn time is ~2 minutes and it should take you at least an hour to get all the skill-ups necessary.

Lockpicking 171 to 228

HORDE / ALLIANCE
Skill range: 171 to 228 
Location: Badlands 
Item to pick:
– Battered Footlocker (Battered Footlocker)
– Dented Footlocker (Dented Footlocker)

HORDE / ALLIANCE
Skill range: 171 to 228 
Location: Desolace 
Item to pick:
– Waterlogged Footlocker (Waterlogged Footlocker)
– Mossy Footlocker (Mossy Footlocker)

TIP: Brace yourself as these are by far the worst skill ups in the process. As chests are located underwater, you can do this either with the help of a Warlock or a Shaman buffing you accordingly, or you could use underwater breathing elixirs. Respawn timer for these is ~5 minutes.

Lockpicking 229 to 250

HORDE / ALLIANCE
Skill range: 229 to 250
Location: Azshara
Item to pick: Mossy Footlocker (Mossy Footlocker)

TIP: You may choose to skip those and save yourself some time as the respawn timer can be deemed too long. You could go to the Slag Pit for an alternative – check the end of this article under Searing Gorge.

HORDE / ALLIANCE
Skill range: 229 to 250
Location: Tanaris
Item to pick: Dented Footlocker (Dented Footlocker)

Lockpicking 251 to 300

HORDE / ALLIANCE
Skill range: 251 to 300
Location: Eastern Plaguelands
Item to pick: Scarlet Footlocker (Scarlet Footlocker)

TIP: This route is accompanied by a video found here (Hyperlink). By the time you finish the route, the first chest has respawned and you can repeat it.

Lockpicking 301 to 350

HORDE / ALLIANCE
Skill range: 301 to 350
Location: Zangarmarsh
Item to pick: Wicker Chest (Wicker Chest)

TIP: These are the easiest skill ups in the process, as the chests are found in abundance and the respawn time is fairly short. An alternative to that is also dungeons such as Steamvaults or Shattered Halls. You could add a pickpocket command to your Garrote macro allowing you to loot a lot of green items from the boxes you steal. May not be the fastest way to level Lockpicking up, but you make more gold in the process.

Alternatives Lockpicking in dungeons

HORDE / ALLIANCE
Skill range: 175 to 250
Location: Scarlet Monastery (26lvl+ Dungeon)  
Item to pick:
– Entrance Doors
– You can use Pick Pocket to steal junkboxes from elite mobs.

HORDE / ALLIANCE
Skill range: 200 to 250
Location: Searing Gorge (The Slag Pit)
Item to pick: Dented Footlocker (Dented Footlocker)

HORDE / ALLIANCE
Skill range: 280 to 300
Location: Black Rock Depths (53lvl+ Dungeon)
Item to pick:
– Shadowforge Gate (Shadowforge Gate)
– East Garrison Door
– The Shadowforge Lock (The Shadowforge Lock)