This dude is playing out the role of Nolan's Joker in real life. He did take on the appearance of the Joker (i. e. dying his hair red). And recall that in The Dark Knight, the Joker's primary aim in causing mayhem is to expose people's hypocrisy.
And that's likely what Mr. Holmes is doing. In fact, the shooting at the theater did expose one kind of hypocrisy. The moviegoers all went to see that film partially because of a desire to watch realistic brutal mass violence (in fact, people seeing films like that often later bitch about the implausibility of the terror scenes), and the shooter gave it to them like never before. So technically, they shouldn't be upset.
In (supposedly) booby trapping his home and daring officers to defuse the bombs there, he's likely trying to expose the hypocrisy of officers who say they care about the public but are actually more interested in finding out whatever cool secrets are in his room. Obviously, if public safety were the only concern, and the bombs were as difficult to defuse as they appear to be, the safest course of action would be to just evacuate the whole apt. complex and let the building blow. But the cops aren't doing that.
The story line reported in the media is that the shooter set a timer to blare music out of his place at midnight to lure police and other residents into getting blown up while he went on his rampage, but the evidence to date doesn't support that line. If that were Mr. Holmes' intent, to simply cause another explosion at the complex to lure police resources away from the theater, all he would have had to do is set up the bombs to explode when his apt. door was opened.
However he ran trip wires all over the place (including lining the windows with them). So that implies that the above wasn't his plan or at least not entirely his plan. Furthermore, if he simply wanted to blow up his complex and kill more people, he wouldn't have told cops about booby-trapping his place when he was caught, as they would most certainly have tried to search his room at some point and tripped the bomb(s), but he did tell them about it, implying he's got another trick up his sleeve.
The supposed bombs in his apt. are probably duds, or else why go thru the trouble of building real bombs only to tell cops later to watch out for them? Also, it's unclear how this dude would've acquired the expertise to build the sophisticated bombs the cops are saying he built--that kind of thing not only requires extensive training but the purchases of the chemicals required could easily be traced. But the sophisticated wiring in the place would certainly lead authorities to believe that they were real, and I believe that's all he wants.
There's a big surprise waiting for cops at that apt.