There is no specific difference in how the new Post 9/11 GI Bill works when you are using it to pay for a doctorate than when you are using it to pay for a lower-level degree. It provides funds based on your tuition, to a maximum equal to the most expensive in-state tuition for an undergraduate degree at a public university in your state. If you are in a graduate program, your tuition is probably going to be higher than that, so you will have to make up the difference in some other way, such as your own money or through scholarships, loans, grants, and other sources.
Many doctoral programs through public universities are fully funded; the tuition is paid for through grants, fellowships, work-study programs, and other means. The way that this works with the new GI Bill depends on how it that funding is given. If your program reduces the tuition to nothing, then you have not need of the GI Bill, since the GI Bill pays an amount equal to the tuition. If the funding is paid to the student, in order to be used for tuition and other expenses, then you can use the GI Bill, and still receive the other funding on top of that.
You’re best advised to speak with a financial adviser at your university, to find out the specifics of how your program will work. That will allow you to make your decision on whether to apply for the GI Bill for that program.