[[WMG: Bill knew Amsterdam was the Priest's son long before the assassination attempt but pretended not to...]]
... because he wanted a son. Amsterdam proved himself a worthy son of both Bill and the Priest in Bill's eye. Maybe Bill saw it fit that he replaced the actual father or that the son of his last worthy enemy was the only person worthy of his "adoption".
* Bill notes everything in the Five Points is under his sway, he probably kept tabs on Amsterdam while he was in Hellgate, he hoped that Amsterdam would come out of Hellgate and do one of two things, either outright challenge him, or, by showing him the ropes and taking Amsterdam under his wing, become his and Priests successor as "king" of Five Points, Amsterdam disappointed Bill by trying to assassinate him, Bill genuinely thought Amsterdam was coming round, which is why Bill let his guard down, and told Amsterdam his backstory, he was hoping to make Amsterdam understand that what he did, was only due to business and rivalry, no actual hatred towards Priest.