I'm going through a similar action myself right now -- I'm trying to kill as few humans as possible. (I made an exception for Benny, and raiders who attack on sight.)
Skills: I know you said you don't want to do locks and computers, but you'll have to for some things, so keep lockpicking and science high. Prioritize speech -- you can get out of a lot of fights, and even better, get XP for it. Maybe go for survival and/or repair, and go after the challenges for crafting.
So some early-game missions to help you out? Skip Primm entirely, head straight to Mojave Outpost. You'll get Ghost's quest there, which just requires you to go to Nipton. In Nipton, you'll get "Cold, Cold Heart," which just requires you to go to Mojave. Easy, non-violent 250 XP. Not much, but it'll get you started.
From there, make a beeline to New Vegas, by way of Novac. In Novac, do Boone's side quest, because it's quick and easy and non-violent. In Freeside, take the peaceful route through G.I. Blues, hook up the Followers and the Garretts, and do the Wang Dang Atomic Tango. Once in New Vegas, go talk to Mr. House and deal with Ring-a-Ding-Ding. After that, you should be able to take down Talent Pool pretty quickly and peacefully.
By the time you get through all this, with perhaps a little bit of minor side adventurism, and you should be at least level 10 or 12.
According to Fallout Wiki:
Mutated animals do not count, only regular animals such as ravens, coyotes, and all types of dog, including NCR, Legion, and Fiend dogs.
Best Answer
To beat that challenge you must kill any animal (not only coyotes but also ravens and dogs). But note that "Mutated animals do not count, only regular animals such as ravens, coyotes, and all types of dog, including NCR, Legion, and Fiend dogs.", as said in Fallout Wiki.