In Harry Potter and the Chamber of Secrets, after Harry kills the Basilisk, one of the Basilisk's fangs becomes impaled in Harry's arm. Tom Riddle, who is still a memory from the diary Horcrux, mocks Harry and tells Harry he is going to die, and that Tom Riddle is going to watch. Fawkes comes along and heals the Basilisk wound with his tears. It appears, then, that Tom Riddle decides to kill Harry himself:
‘Phoenix tears ...’ said Riddle quietly, staring at Harry’s arm. ‘Of course ... healing powers ... I forgot ...’
He looked into Harry’s face. ‘But it makes no difference. In fact, I prefer it this way. Just you and me, Harry Potter ... you and me ...’
He raised the wand.
Chamber of Secrets - page 237 - Bloomsbury - chapter seventeen, The Heir of Slytherin
In chapter fourteen of Goblet of Fire, The Unforgivable Curses, Barty Crouch Jr as Mad-Eye Moody tells the fourth years that Avada Kedavra is a curse that requires "a powerful bit of magic behind it."
How could Tom Riddle, a memory, perform Avada Kedavra?¹ Yes, he uses Harry's wand to rearrange Tom Marvolo Riddle to I am Lord Voldemort earlier in the chapter, but there's a huge difference between rearranging letters and Avada Kedavra. Can a memory hold magic powerful enough to cast Avada Kedavra?
¹ The implication is Avada Kedavra. I seriously doubt Tom Riddle was about to cast Tarantallegra.