After Grimmjow was defeated by Ichigo, Nnoitra Gilga, the 5th Espada, barged in and attacked Grimmjow. Ichigo defended him from a further attack. That is the last time Grimmjow was shown and he was alive, breathing and talking at the time. While it is possible that he has died since then, there is nothing in the story to suggest that he did. So as far as we know, he is still alive.

Kubo has since made comments about his plans for Grimmjow's death early in the Arrancar arc, but he changed his mind and never confirmed or denied if Grimmjow was alive. There are people out there who misunderstand what he said and spread falsehoods based on that misunderstanding, hence some confusion. As the above says, he may be dead but based on what we have been shown he was last seen alive.

In an interview during late January 2012, Kubo finally revealed that Grimmjow is indeed alive.