Haytham Kenway was never an Assassin. His father, Master Assassin Edward Kenway, had begun to instruct him in sword fighting and the basic philosophies of the Assassin Order when Haytham was a child. However, before he could tell Haytham why he was being trained, Edward was murdered by mercenaries hired by Reginald Birch, a business associate of Edward. Birch, unbeknownst to Edward, was the Grand Master of the British Rite of the Templar Order. Haytham's sister, Jennifer, was kidnapped and Birch secretly had her sold to slavers. After that, Haytham studied under Birch as a Templar.
Many years later, Haytham had been tasked with retrieving a manboy names Lucio, whom was being guarded by an Assassin name Miko. Haytham managed to kidnap Lucio, but ran into Miko shortly after while descending a cliff. Needless to say, a fight ensued, during which Haytham's short sword was lost. Haytham managed to knock Miko off the ledge, but Miko grabbed onto his arm. Haytham proceeded to remove Miko's hidden blade in an attempt to get him to lose his grip, which he succeeded in doing. Miko survived the fall by grabbing onto the rope Haytham had been using to climb down the cliff. Haytham gave Miko the choice of falling to his death, as he was poised to cut the rope, or leaving and living to fight another day. Miko decided on the latter option, and Haytham kept his hidden blade as a replacement for his sword.
Also, just to key you in on the irony of the situation, the first person Haytham killed with that hidden blade was it's former owner, Miko (The man in the theatre box in the opening sequence of AC III).