The card text states:

"Any Phase. Bring Jaqen H'ghar out of shadows…" and attach him to a non-Army character in an opponent's dead pile that was killed this phase.

My question is how can a character be killed this phase (before cards come out of shadows) if Jaqen H'ghar has to come out of shadows at the beginning of the phase?

Because he doesn't have to come out of Shadows at the beginning of the phase. His "Any Phase" ability gives him the ability to come out of shadows during any action window.

radiskull said:

This.

It's the same way you can use an any phase action to bring Meera Reed out of shadows to blank a card (if you bring her out during the actual shadow mechani window, she doesn't work)