There are A LOT of PhD programs where you can focus on computational chemistry (though, a lot of them are just lumped into the Physical Chemistry section). And yes, DFT is a big part of it.
There are people who develop methods, and those who just use it to find structures, etc.
Welcome to the hard part of finding a PhD program to go to. So, start with big good schools, and even look at regions you might want to live, visit the websites of the schools and read about the professors. Email the professors you think you might want to work with and tell them what you are interested in, when you plan on applying, if they might be accepting students then,etc.
This process, picking the right schools to apply to, takes a lot of hard work and time. And apply to as many as you can afford, and I would recommend only applying to schools unless there is at least 2 people you think you would be willing to work with.
Then visit all the schools you are accepted to (the good schools pay you to visit).
Good luck!