How do I install a vent above a closet door?
Depending on the structure, the header of the door may be load bearing and you'll be met with a solid block of wood behind the drywall that shouldn't be cut. Drill a small test hole from the inside of the closet first.

Otherwise, installing a return grill on each side of the wall is fairly simple. Leave at least an inch or two border to the edge of the return grill so that the screws are not near the edge of the drywall. And it wouldn't hurt to use some plastic drywall anchors because drywall itself will easily crumble if you over-tighten the screw.

A few tips, turn the grill so the vents face the ceiling where you cannot see into the hole. And if you can, make a second vent towards the floor since you need a source of cooler air to replace the hot air you're trying to vent out.