OP asked for the "most independent class," not the best soloist. Arguably that might apply, but what if "most independent" means other things, like not needing gear, not needing to buy supplies, not needing to go to town to buy often, etc.
Mages summon their own food and water, free. But I find mages to be somewhat gear dependent in that they suffer w/out an enhanced mana and sta pool.
Hunters are independent in terms of being great soloists, but they are very dependent on ammo. If you are a bow hunter, you must go to the store to buy new arrows. If you are a gun hunter, you can avoid that by having engineering skill and making bullets from stuff you mine - provided the nodes to mine are in the area you are hunting -- and you are still dependent on that. Hunters need a decent range weapon, an okay melee weapon - but don't strictly NEED great armor to pve.
Warlocks might be the "most independent" in that they can tap for mana and drain back health from targets while fighting behind a taunting VW pet. They are great soloists and have no gear requirements (all meleers are gear dependent). A warlock can level to 60 in his underwear, fighting behind a taunting vw pet. A hunter can do the same, except he has some dependence on weapons and ammo.
All classes can solo, if that's what you mean by "independent." All.
Some level faster than others - but not if you hate doing it. Mages can AoE for fast experience (harder than it sounds, btw). Rogues are very fast killers but are gear dependent. Hunters and locks have good to great dps and kill very fast as soloists. Beware holy spec priest, healing spec druid, all paladin specs (okay, maybe not retri spec pals, but even they aren't "fast" killers) and heal spec'd shamans - and tank-spec'd warrior. These can level very slowly compare to others.