Well, here's what I think brother, done this alot and I think this is going to be your best option. You'll buy everything besides the mylar and light at hardware and fabric stores.
Prep: Remove your doors, the closet rack, carpet optional. Buy a roll of mylar, it should cover the entire closet, floor and ceiling included. If you're concerned about the floor, spray paint a tarp with waterproof sealant that is flat white and attach it to the walls a small way up the wall so the water doesn't run onto the floor. An inch will do. Put a 2x4 across the floor in front of the closet and attach the tarp here as well.
#1. Door. Given this looks like a space thats going to have light in the room outside of it it and this is your flowering area you need a solution. Go to a fabric store. You're looking for something called "blackout fabric" - its professional grade light blocking shit, photographers, people on night shifts, etc use it. This will stop all light completely (I have a 600W MH behind this fabric and there is zero glow) and its flat white and will reflect light well. Buy a roll of velcro (also sold at the fabric store), it will be double sided sticky, attach it to your fabric and doorframe and you will have an easy access curtain that is lightproof. Fabric is usually $7 for 3 feet and comes 54' tall standard. The velcro will vary but its cheap. If you would like to line side facing into the grow space with mylar, by all means go for it, just glue that shit together.
#2. Lighting. A 400 watt will do just fine for that growspace. If you are unsure for mounting, get a stud finder and an eye bolt. Its a big screw that has a circle on the end
. Mount the eye bolt in the center of the closet IN A STUD and hook chains from the eye hook to your reflector. If you have a better method of hanging a reflector go for it, this is just what I use in closet setups. You may aircool the hood if you like but a 400 in that space should be fine heatwise if you properly ventilate it.
#3 Venting. Calculate the total cubic footage of your closet and get a fan thats in the ballpark but on the larger side than what you need. You need to replace all the air in 5 minutes. Every fan will have a CFM listed, this stands for cubic feet per minute. Mount the fan inside the closet and buy some ducting for the exhaust, you can cut a hole in the fabric with an xacto knife or a razor blade, put the ducting through it and seal with duct tape. For intake buy a dryer exhaust cap, cut a hole, and mount it into the fabric baffles facing so they will open inward. The negative pressure created in the closet by the fan will draw enough air through the intake with no problems, laws of physics demand it, its not the power of the fan, rather the power of the atmospheric pressure. If the baffles do not lie flat, tape something to the back of the baffles so they fall shut during the fan's off cycles but not heavy enough that the baffles do not easily open during the on cycle. You'll have to play with this.
#4. Plants. . . . . this depends on how you're going to grow your plants. However they fit should be fine, light will fill that closet very well if you do the mylar. All thats left is to monitor heat and humidity and adjust your fan accordingly until you're at where you want to be.
Well man, good luck and happy growing! If this helps hit me with some rep. I needs it.