turbo size is gonna depend on what motor you run, how high you are going to spin it, and how much power you want to make. I have a truck 6.0 sitting in my GN right now, I'm going to run the stock longblock, maybe change the cam and rod bolts, and that's it. It will get around an 80mm turbo, which is pretty small for 364ci, but it's a street car.
You could run an 88, 91, or 94mm on these motors pretty easily, just depends on what you want to do with it.
As far as 6k, it's probably going to end up costing more than that. Depends on how much of it you do yourself. I bought my 6.0 for $750, 4.8s and 5.3s can be found even cheaper. You could possibly do it all for less than 6k, but keep in mind your transmission, fuel system, mounting, and turbo hot side costs.
I bought EFI live from a member here, that's what I am going to tune mine with.