Unfortunately, the method for this is pretty shabby and typical. You can hack out a fair complement of attacks with DJ's own scythe, either with a button or just by tapping enemies - however, you still have to point him toward the beast with the D-pad for the attack to connect and the tapping timing is sluggish, so it doesn't work as well as you'd think. Over time, you'll unlock a few more weapons, like pistols, a shotgun, and suicidal hamsters with C4 strapped to their backs (they can open new paths as well). Ammo is pretty limited though, so it's usually the scythe or nothing.
Unfortunately, combat is the good part; you'd think the ability to double-jump, combined with the scythe's power to propeller glide, to grab onto high ledges or to swing from hanging hooks would make DJ a veritable Spider-Man. It doesn't. You're mostly hampered by swooshy controls, but the camera views and the fact that the scythe doesn't grab that ledge you'd hoped it would at least half the time also contribute to the anti-awesome.
There's a whole second system in the game, in which you can take control of Pandora's spirit and navigate the spirit realm for 15 or so seconds at a time. She can sometimes reach areas DJ can't (spirit platforms, you see). But mostly she collects the souls of fallen enemies and then shoots them, slingshot-style, to replenish DJ's health and confidence (a mostly useless meter that helps you do more damage when you're feeling potent), hit spirit bull's-eyes that figure into puzzles, and stun enemies.