This boss has a fixed battle pattern no matter what mode you're on. Follow that pattern and you're guaranteed victory.

Doppelganger takes damage only when he's vulnerable during the time he's exposed to light. So this is the way to trick him into the light:

There are ‘panels' (I don't know what to call them) on the walls around the battlefield. When those panels are hit a certain fixed number of times, light will expose. Below lists the number of times to hit them before the light exposes:

(Based on DMC3 Special Edition difficulty modes)
Normal: 3
Hard: 5
Very Hard: 6
Dante Must Die: 7

1. The plan is to hit the panel one time less than the number listed above. For example, on Normal mode, hit the light 4 times first and stay there right by it. Shortly, Doppelganger will rush in for an attack.
    
2. Once he gets close enough, hit the panel one last time and the light will expose on him. That is when you attack. Timing is important so try getting used to it to avoid mistakes. Doppelganger doesn't stay vulnerable for long, so you'll need to repeat this procedure until you finish his life.
    
3. The fastest way to kill him is to use activate Time Lag (Quicksilver) the moment he's vulnerable and spam Beowulf on him. (He's weak against the element of light) For fastest speed and best damage, I recommend jump canceling Killer Bee.
    
   While this damages him a lot and fast, it's not enough to kill him in a single opportunity - You'll need to stock up more DT to repeat this. The fastest way to regain DT is to taunt, and the safest time to taunt is 1) when he backs away when a panel is broken but the light doesn't reach him 2) after he finishes attacking. After taunting once, you'll need to hit at least one panel before you can taunt him again for DT. The rest is rince and repeat.
    
4. Doppelganger does a ground blast attack from time to time. Fortunately, it is rather slow and easy to avoid; just keep running.
     
5. Note: Exposing ALL the panels will turn Doppelganger vulnerable for a longer time. Doppelganger may shut the panels down by force (using a black orb) one by one when he sees that most have been exposed. However, it is possible for you to deflect the black orb back at him with a melee attack. This will save the panel from being shut and damage him at the same time. In the harder modes, though, Doppelganger learns to deflect it back at you too so you may have to deflect it a few times before it finally hits him.

Hard mode - Old video demo with text (2.29 MB)
This video demo does not reflect the updated strategy as written above.