Rick-
In a strong, modern revolver, the 45 Colt is all the handgun you'll ever need to hunt deer, elk, black bear, or what have you. It used to be that only handloaders could take full advantage of it, but now Buffalo Bore and others are offering what you might call "+P" loads for the 45 Colt. These loads are strictly for Ruger, Freedom Arms, T/C type handguns, not Peacemakers or their clones. I've used AA#9 or H110 under 255 grain lead bullets in handloads for a Ruger Blackhawk, and it's a great round to handload.
Even if you hunt with a Colt Peacemaker, there's no reason why it can't take whitetail deer out to whatever range you're up to. In a 7.5" barrel, the original black powder load sent a 255 grain bullet out the muzzle at over 900 fps. That will drop any deer around, if it's well placed. Several loading manuals can give you handloads that safely match or approach that original power level in Colts and Colt clones.
I wouldn't use some of the current down-loaded cowboy action shooting ammo, since it's about 200 fps slower than the original black powder loads, and some of it is loaded with light weight bullets, too.