I am level 60 skinning but am level 15 with my overall account and want to know where it is suggested I level up my skinning the quickest? 

I also am a skinner.And a undead for me I found the best and quickest level in skinning is in the Barrens/thousand needles. Both are contested territory but rarely have any problems there. but if you are ALLI then I would say there is great skinning in Ashenvale.

there is no way too lvl your skinning level faster..

just sking monster whit orange skinning and u will have your skinning skill up real fast ...

calcul is done like this...

your skinning lvl divide by 5 = max level you can skinng
heinouca lvl 70 hunter
375/375 LW 375/375 skinner
serveur : Exodar

The previous comment is correct when stating that skinning is easy and can be leveled quickly. However, I want to add to that, so you don't have confusion once you pass lvl 150 in skinning.
Once you get to 150 in skinning, you will begin noticing that you do NOT gain a level of skinning each time you skin,,,even if the skinning attempt is "orange". This gets worse, too...Once you get around 300 or so, it will have gradually worsened to 1 level-up every 5 to 7 skin attempts. Therefore, when you see this start to happen, don't think there is an issue in the game or you are doing something wrong. It's in the software coding of the game. With that said, skinning is still very easy to level up. Have fun!

im a hunter to me the best place to skin is at the Arathi Highlands start farming the Raptors they got stuff that sell really good at the vendors and u can also skin the raptors so it hit two birds wit one stone and if ur just starting out in arathi highland then its also great to lvl too