A closet (your standard 3x4 or 4x6 type or small walk-in) is not and will not make a usable vocal booth without complete reconstruction.
"Egg crate" and foam are fairly worthless (as they don't do anything for the problem frequencies that are absolutely HUGE and out of control in a space that small - not to mention that the problem frequencies are actually inside the fundamentals of the baritone range - It's a really, really bad idea).
The only thing that *might* make it usable (no guarantees) is ripping out all of the interior drywall and filling the studs with something along the lines of OC703 or 8# rockwool. Then you can frame out cloth frames to cover the studs (to control fiber migration).
Again, no gurantees that will *fix* the problem, but it might come close.
Generally, the best thing you can do with a closet is us it as a place to store noisy equipment.
