Of course he has the authority to fire Comey. That is not the issue. The reason Trump himself gave, discounting his own administration's excuses mind you, was that he was thinking of "this Russia thing" when he fired him.

To be honest I don't think Trump has the mental capacity to have actively colluded with the Russians to hack the election. I also think he has lived such an unaccountable and privileged life that he actually believed that he could just ask the FBI director for loyalty and to drop an investigation, and then fire the same director for disobedience without anyone questioning his motives. 

I think the reason Trump is so chummy with the Russians is because he has been laundering money for the Oligarchs for years. This is what he is afraid will come to light, not any form of election collusion.