How can I play as Judas?

You can unlock Judas by first beating Mom, which will unlock two new dungeon levels called "The Womb". Then you have to beat The Womb, by killing the boss Mom's heart.

There seems to be an answer to this in the Steam achievements of the game. First, lets look at a couple of achievemnts that seem to support this guess.

Maggys Wig Collect 7+ container hearts in one play through
This is the requirement to unlock Magdaiene. (As I recall it. Could have been less.)

Cains Eyepatch Collect 50+ pennies in one play through
This is almost the requirement to unlock Cain. (The actual number is 55.)

Thus, I feel it's reasonable to assume that the requirement to unlock Judas is connected to the achievement

Judas's Fez Complete the full game

What exactly this means, I do not know, as I haven't yet beaten Mom.

And looking at the achievements it seems that just beating Mom isn't enough.