There are quite a few Caribbean locations with nice diving that can be done for less than $5k. $5k will get you a week pretty much anywhere unless you really blow it out on your accommodations. But are any Caribbean locations "great" when compared with top notch Pacific locales?
In the Caribbean, my top pick is Bonaire. I just got back from my third trip-- $700pp r/t Atlanta to Bonaire, $1200/wk for a two bedroom condo on the water with great diving right in front. $350/wk for a small SUV. $175pp unlimited tank fills for a week. You can manage food costs how ever you want. There are restaurants and every level, and with a condo you can eat in if you want. Bonaire is the easiest diving there is and because it can all be done from shore, you are free to dive your own profiles on your own schedule. You can also go to a resort (Captain Don's) if you want more structure.
I've also loved St Croix (Tamarind Reef, diving at Cane Bay and the Fredrickstead side), Little Cayman (LCBR) and St Lucia (Ti Kaye, really nice diving in the marine park), and would go back. But Bonaire is my top pick, especially if you have someone new to diving.
My experience in Cozumel (three trips) is that while it is definitely inexpensive, the currents can be incredibly strong, even dangerously so, and it is less cheap that it used to be. I've also been to Ambergris Cay in Belize, which I enjoyed, but I wouldn't go back over the others I've listed, and St. Kitts, which I didn't much enjoy. Also Grand Cayman (two trips), but I wouldn't go there again either. Diving on the East End is nice, but you are better off taking the hopper flight over to Little Cayman.