I don't know what your local laws are, but some states don't allow .22 centerfires for deer hunting. Texas does, but some states don't. If you are a serious deer hunter, I would recommend something a little larger, like a .243 or 25-06. I just got a 25-06, and plan on carrying it when I'm hunting deer in more open country. I'll still use my 30-30 for stand hunting and in brushy areas where shots are close. I may also carry it when hunting coyotes on open ground. The much larger .25 caliber bullets buck wind a lot better than the .22 caliber bullets. The only downside is they damage the pelts a little more. Many coyotes in my neck of the woods don't have good pelts, so I don't care.

A .223 is capable of taking a deer, but I would limit my shots to under 100 yards. I would also aim for the neck. The thing that limits any .22 caliber rifle is bullet selection. Most factory ammo is made with thin jackets for rapid fragmentation. This makes neck shots deadly, as bullet fragments tend to hit everything. However, on a chest shot, the varmint bullets are seriously handicapped because of their thin design.

As far as .223 or 22-250, the 22-250 has an advantage in ballistics under ideal conditions. But the .223 has a lot better selection of ammo and is a little cheaper to shoot. I got a .223, mainly because I wanted a gun that I could afford to do a lot of shooting with. And the wind here usually causes enough drift to take away the added range of a 22-250.