Apparently he did dye his hair red, like the Joker. He bought a ticket and went in as a normal patron. After some 20 minutes into the film, he pretended to get a phone call on his cell. He exited an emergencey exit where he must have stached his commando gear earlier, then somehow reentered. It might be that he proppe the door open or knocked and someone openned it for him. He throws tear gas cannisters into the room, then goes up and down the aisles, firing away. Many people were shot multiple times. The wounds were horrific.
I guess he just walked out when the police entered and surrendered.