I will repeat the above information. Nonresidents *must* hire a guide if hunting in a wilderness area in Wyoming. That said, there are plenty of places where you could hunt on a general tag without going into the wilderness areas. Mostly look at western Wyoming, although there are elk in a variety of places across the state. Look at the Bridger Teton National Forest, and the Shoshone National Forest for starters. Look at the info provided by the Wyoming Game and Fish to see where the wilderness areas are at, which you must avoid if you do not have a guide.
Grizzly bears are an issue across both forests. By yourself, the bears may be more of a problem. You have to be careful of how you store your food, following forest regulations. Bear spray is a good idea too. You might also consider a SPOT system, or something like that, in case you get seriously injured. Again, I am assuming you are alone.
Without a guide, you are not going to be hiking too far back into the mountains, because of the wilderness area boundaries. So you could easily imagine a base camp not far from a road perhaps, and then hunt around and up to the edges of the wilderness areas. The elk hunting is not necessarily better in the wilderness areas, but those areas do usually have less hunter traffic.
If you are hunting alone, and you kill an elk, you will probably end up leaving some of the elk on the ground overnight. That is where the grizzly bears show up. Plan for that, and be extremely careful on your approach the next day.
If you hunt early in the season, and it is warm, you will be in a race to get your meat out and cooled down. Doing it on your back is hard. Horses are easier if you have them.
You could find many places to hunt around and in the area defined roughly by a triangle from Cody to Dubois to Jackson, or even a bit farther south. Try calling and talking to the biologists in the local Wyoming Game and Fish offices in whatever area you are interested in. Elk are where you find them. But try to get away from the towns and the highways. Weather has a very strong influence on where they will be. Also on where you will be.
If you are on foot, and hunting by yourself in an area that you do not know, plan on extra time to try to get an idea of where the elk might be. If you have hunted elk before, it will be easier to figure this out.
Also your DIY elk hunt will be a lot more fun if you are in good physical shape, and you have some well broken in boots.
I am totally in favor of a DIY hunt, but I must say that for someone who does not know the state, it would be very much easier to hire an outfitter. You might spend your precious hunting time just hiking around and learning the area. By the time you start to get some of it figured out, your hunting time might be gone.
Wyoming is a fun place to hunt elk, but be realistic about your chances of getting one if doing this on your own. It is very possible to do so, and hunters do it every year, but the elk might not be where you are hunting on that day.
Wolves are also an issue. If a wolf pack has moved into the area that you just chose for your hunting camp, you are probably better off to find somewhere else. Ask the biologists about that, if you manage to get a tag.
Good Luck.
Wyowind