As the others have stated! Yes a 223/5.56 will kill a deer! However being an avid deer, pig and coyote hunter I would not recommend a 223/5.56 for deer hunting. The smallest caliber I would even consider for deer would be a 243 Winchester. But better yet is the 308 or 270 Cal. would be excellent choices. You could also use lighter bullets in them for them to varmint hunt. But both would make excellent deer rifle calibers and there are rifles that are very good and reasonably priced. Like the Mossberg, Marlin Savage, Howa, and other good bolt action rifles. Of course if you hunt thick woods country the Lever Action 30-30 rifles are good for deer. I prefer the Marlin 336 Models for this since they are easy to mount scope mounts and a scope on. But the 30-30 is not the best for open country and longer shots.
Also do not forget to look at top rated used rifles. Like Remington, Winchester and others! The reason most buy a hunting rifle with good intentions and dreams of going hunting. They sight the rifle in and shoot a few rounds and the rifle sets in a safe year after year! And they are a great deal regarding price! They technically are still used! Good luck on your selection.