Yes, you can rent a car in Playa del Carmen and do Tulum on your own, however,play on spending a bit of time getting your rental car squared away (and be very careful that you check your car over thoroughly), then about a 45 minute drive to Tulum, several hours to see the site (suggest hiring a guide unless you are well-versed on Mayan history and architecture), then drive back and more time to return rental car. Next you will need to catch the ferry, which generally runs every hour, back to Cozumel.
If this sounds like too much work, spend your day in Cozumel. You can rent a car here and spend the day touring the island and stopping along the way as the spirit moves you. While Chankanaab is an option, we who live here don't consider the snorkeling to be all that great and usually head for Dzul Ha instead.
Ross in Cozumel