The hidden blade with the Assassin Symbol on it was a stolen blade from Miko (the man Haytham kills at the beginning) to supposedly replace his own broken sword (since the novel by Oliver Bowden is his own ideas on the AC characters, it's hard to know how much of it is canon), where the other blade (he has two)
came from is a mystery.
Haytham was being trained to become an assassin from an early age by his father, Edward (AC4 Black Flag's protagonist), much like Federico was by his father, Giovanni, in AC2. Much like Federico and Ezio too, Haytham didn't know why or of his heritage or the Templars being enemies etc. only his daughter, Jennifer, knew what her father's true vocation was, so his mother wasn't privy to this either.
When Edward discovered that Reginald Birch was a Templar (his daughter told him) he argued with him and told him to get out, however Haytham had inadvertently told him about his father's secret safe in the library as that was where he had put his son's future sword and he was after the journal documenting first civ stuff etc. He got Braddock's thugs (in disguise of course, hiding the Templar symbol on them) to break in to try to retrieve it where a fight broke out which ended up in Edward's death and Jenny was kidnapped and sold into prostitution. Haytham defended his mother to save her from the same fate as Maria Auditore (rape or sexual assault) by the thugs by stabbing one of them in the eye, killing him instantly, and she stopped loving him for it forever - she handed him over to Birch and she died years later having never forgiven her son.
He was thus indoctrinated into the Templar order by Birch (from the age of 10) and didn't find out until later having found Jenny that he had been lied to and Birch had been responsible for his father's death and his mother's disowning of him too, though why he would remain a Templar is beyond me. Although in AC: Unity we find out he wrote letters to his sister (who was in contact with Elise and her father who have so far been the only nice Templars in the series) saying he wanted to unite the Templar order with the Assassin order (perhaps after meeting his only son, Connor maybe). Pity both Charles Lee and Achilles got in the way of his relationship with his son, Achilles just wanted revenge for Haytham crippling him and used Connor to do that, and Charles had become a corrupt power hungry drunk and wanted Connor dead and not just because he was an Assassin but because he thought he was too much of a distraction for Haytham and wanted Haytham to kill him - which he didn't do because he was his son and loved him (that piece of crap, Otso Berg said Haytham was weak for loving his son and should have killed him regardless). So Charles manipulated Connor's childhood friend instead, which unfortunately backfired and made Connor want Lee dead even more. It is a shame that life for Haytham had been so different to what it should have been, Edward would have never wanted this for Haytham although I do think he should have been less clandestine with his family especially his wife and son with regards to his heritage - sad really.
PS: It is the Animus that fools the player (and Ubisoft, of course) into thinking Haytham is an Assassin not Haytham himself.