Alexander was born in 2366 to Federation Ambassador K'Ehleyr, a half human-half klingon woman who was a lover of Worf twice in the past. She did not tell Worf about their son and Alexander spent his first two years with her. In 2368, K'Ehleyr was killed by Duras and she died in front of Alexander in Worf's arms. He told him to look at her face, the face of death, and always remember. After her death, Alexander returned to Earth to live with Worf's adopted parents, Helena and Sergey Rozhenko. At first Worf would not recognize Alexander as his son because he did not want his discommendation to bring him shame.

In 2372, Alexander and Worf were visited by a Klingon calling himself K'mtar saves Worf and Alexander from an attack on Maranga IV at the Kot'baval festival that Worf had brought Alexander to in order to convince him to become a warrior and embrace his Klingon heritage. K'mtar tries to help Worf train Alexander as a warrior and when they fail and Worf refuses to send his son to a Klingon training academy, K'mtar threatens to invoke ya'nora kor. Worf later finds K'mtar in his quarters preparing to kill Alexander and attacks him. K'htar reveals that he is Alexander and has come from the future to change his own future. This meeting convinces Worf to change his attitude towards Alexander's upbringing and convinces him to let his son follow his own heart.

In 2364, he took part in the wedding ceremony between his father and Jadzia Dax. As Alexander grew up he resented Worf for leaving him with his grandparents as a child and he enlisted in the Klingon Defense Force. He and Worf served together aboard the I.K.S. Rotarran, but tension between the two of the made him decide to transfer to the I.K.S. Ya'Vang.

Sources:
http://www.startrek.com/library/individ.asp?ID=114489