To trigger X-Mode, you must have a matching suit and X-Gene set. Hit and equip them all to match a specific Marvel character. Once you do so, you then must have at least 4 bars of M-Power. This is the blue bar located below your health. Once you have these specific things met, you can trigger the power. Just press to do so.
Suit: This is the piece that shows what the X-Mode power will do.
Offensive X-Gene: This piece will usually add status effects to attacks, among other things.
Defensive X-Gene: Usually associated with dodge or M-Power bar being filled for extra defense buffs.
Utility X-Gene: Gains more M-Power quickly, adds an extra bar of M-Power, or gives you a way to move faster in the game (by holding and ).
Note: You may not get this for a long time depending on your luck. All of the X-Genes and Suits are random, so it's completely based on chance.