I think that the premise went like this:
Ginny has been writing on the diary for quite a while. Turns out she's been writing, all the time, about how the famous Harry Potter defeated the Dark Lord - as a baby.
If the baby Harry destroyed Voldemort, knowing this, would Tom Riddle dare try again, without first figuring out how exactly did it happen? He's not stupid enough to commit the same mistake twice... ok yes, yes he is, but this time it is pretty obvious that it would be overly imprudent to attack someone who somehow destroyed his adult version with no reasonable explanation.
- (from Chamber of Secrets - Chapter 17 - The Heir of Slytherin)
"I have many questions for you, Harry Potter". "Like what?" Harry spat, fists still clenched. "Well," said Riddle smiling pleasantly, "how is it that you -- a skinny boy with no extraordinary magical talent -- managed to defeat the greatest wizard of all time? How did you escape with nothing but a scar, while Lord Voldemort's powers were destroyed?" There was an odd gleam in his hungry eyes now. "Why do you care how I escaped?" said Harry slowly.
"Voldemort" said Riddle softly, "is my past, present, and future, Harry Potter..."
"To business, Harry," said Riddle, still smiling broadly. "Twice -- in your past, in my future -- we have met. And twice I failed to kill you. How did you survive? Tell me everything. The longer you talk," he added softly," the longer you stay alive."
On the other hand, asking the Basilisk - a third party - to kill Harry seems to be safest bet. If Harry has some sort of miraculous defence mechanism, it will be the Basilisk that ends up dead, rather than Tom Riddle.
So Harry kills the Basilisk. Why didn't Tom, on a desperate move, attempt to kill Harry? Because Harry was already doomed anyway: Tom knew that Harry was poisoned and it was just a matter of time for him to die, so there is no need to cast a spell that may or may not have adverse effects on Tom.
So, the safest bet is to simply wait for Harry to die off from the poison...
... aaaand Fawkes arrives.
Riddle's face contorted. Then he forced it into an awful smile. "So. Your mother died to save you. Yes, that's a powerful countercharm. I can see now... "