What caliber of gun kills the most people each year?

First, guns do not kill people - they are inanimate objects. Probably the most common firearm, and due to commonality, likely to be used in more shootings, is the .22 rimfire.

Overall, based on sheer numbers produced and the intended audience, the AK family of automatic assault rifles likely kills the most people world-wide. The AK family is generally chambered for one of two calibers: the 7.62x39 or 5.45x39, with the 7.62x39 being far more common (about 10:1 in numbers).

If one is talking only about murders and other deaths in a civilian setting (not wartime or other settings), then the most common would likely be one of: 7.62x39 (AK family), 9mmx19 Parabellum (most handguns), or the .40 S&W (majority of police handguns). Note that, world-wide, suicides and gun accidents make up the majority of gun-related deaths in a civilian setting, with murders and police shootings coming in well below either of them.