Goku's character is unique that way: he's the psychopathic blood knight that constantly seeks out stronger enemies to the point where it's self-destructive. He also just happens to be a nice guy as well. He also has a bit of a child-like naivety about how inherently good people are (they're not).
He's basically Vegeta in the "Saiyan must have good fight" but the difference is that Goku likes challenges for fun, where as Vegeta likes it for ego. Hence his speech about GOku in the Kid Buu arc.
He didn't have a chance to let Cell live. He died because Cell was going to blow himself and Earth up. Besides, he was already beaten by Gohan. So Gohan would have been the challenge Goku would seek.
Goku was telling Gohan to kill Cell straight away, but Gohan refused.
I don't blame Goku though, Cell was created for the sole purpose of killing Goku, I'd want his ass dead too. The Androids were made for this purpose too, but they are part human, and have some humanity to them.