yes a 223 can kill a deer if shot placement is observed and using the proper ammo. but IMO also, 243 is the smallest caliber i would be comfortable using for deer size game.
need to check your states game regulations as some states have a minimum size caliber that can be used legally. even a centerfire 22 caliber is illegal in many states. need to to check this first before making any decision on caliber.
check out the Marlin X7 series and the Savage Axis series of rifles if considering a new bolt action rifle for hunting. also check out the pawn shops and gun stores for good used rifles. many good decent hunting rifles can be found and many can be had for a half to a third the cost of the same rifle new. sometimes they may even have a decent scope on them.
my suggestions on caliber for deer sized game, would be these as they are going to be the most commonly available. 243 Win., 30-30, 30-06, 270 Win., 7mm Rem. Mag., or 308. any of these wil work just fine as long as you use proper ammo for the job.