A "searching look" generally means to look at someone as if trying to discover additional, unspoken meanings to something they said, or additional, unspoken reasons for why they said it.
So in this context the captain is looking carefully at Alex to discern whether Alex is concealing information, and if he might see some clue to what it might be. Perhaps Alex knows what happened to the previous crew. Perhaps Alex actually did something to that crew. Perhaps it's weird that Alex offered a reward for finding out, and the captain wants to try to figure out Alex's motive for doing so.
The response, "Don't worry we will", is a little odd, though. I suspect there is either more to the story, or some kind of mistranslation.
Before the two men entered into the house, John pulled Ben aside. "Ben," he began. "I just ... Look. If you see something weird in there, promise me you won't freak out, OK?"
Ben paused and gave John a long, searching look. "Well, now you're already making me freak out, man. Is there something you're not telling me about this place?"