Nothing can take away my desire for revenge:

When Archangel moves at least 5 squares in a direct path, he can use Sidestep and Energy Shield/Deflection until your next turn.

Archangel has two clicks of Hypersonic Speed middial. Assume he initiates Hypersonic Speed, moves 5 squares in a direct path, attacks, and then moves in some random direction or directions, whether a direct path or not.

Does "Nothing can take away my desire for revenge" trigger or not?

I'm thinking it does, but I'm asking because, quite frankly, this feels a bit busted.

Yes he would gain Sidestep and Energy Shield/Deflection. His ability only cares and triggers if he moves at least 5 squares in a direct line within a movement. As long as you meet that requirement, you get the benefits.

I imagine this would also trigger if he moves 5 squares in 1 direction, and then the 6th square in a different direction and then attacks. (like for his charge)

In order to qualify as a direct path, the entire movement from start to end must be straight and in one direction. For HSS, it's 2 distinct movements, so at least one of them would need to be direct.

This. It's covered on page 11 in the rulebook. Emphasis is mine.

DIRECT PATHS
When a character moves in the same direction (horizontally, vertically, or along a perfect diagonal) without changing that direction for the entirety of that move, it is moving in a direct path. A character moving in a direct path can’t move backwards along that path and must move a minimum of one square.

Right.

Moving 5 squares in a straight line and then 1 square not in that same straight line is not moving in a direct path, and he is looking for two things: Did he move 5+ squares? Was that movement in a direct path?

With HSS he could move 5+ squares in a direct path as either of his two allowed movements and trigger the effect, though, regardless of what he did with the other allowed movement.