Voldemort wanted to kill Harry because of a prophecy that Severus Snape overheard before Harry's birth (told by Professor Sybill Trelawney to Albus Percival Wulfric Brian Dumbledore). The prophecy said that "The one with the power to vanquish the Dark Lord approaches..." Snape told Voldemort of the prophecy, and Voldemort decided (most unwisely) to find the person who was mentioned in the prophecy and kill them. He decided that Harry Potter fit the prophecy ("Born to those who have thrice defied him...born as the seventh month dies..."), so he set off to kill young Harry.