It would be logical since that happens in every movie/tv show

Add some random time required to pass to turn a dead character into a zombie
the time should depend on characters state when he was alive, for example if he was hungry, thirsty, or was bleeding he/she should "turn" faster

Add that if you shoot a dead (or a live) character in the head that he does Not turn into a zombie

Also it would be nice if zombies would crawl if their legs were broken (if a character died with broken legs)
if a zombie could not attack i it has broken arms, but could still follow you and make noise
if you could add that you could brake zombies arms and legs as it can be done to a character