Buy a faster printer.
And, no, I'm not being a smart-aleck
HP rates this printer at up to 32 pages per minute -- in fast draft mode black ink only... But even their ratings show it takes over a minute per page when doing quality printing in color.
And those speed ratings are all "after first page" -- ALL printers take longer for the first page due to the time it takes for the computer to set up the page and process the information.
As the specs suggest, you can get faster printing in black-only instead of color and by choosing fast draft mode instead of quality or normal mode.
There is very little that could be wrong with the printer that would make it print slower than usual -- generally any problem would make it not print at all or cause it to print incorrectly. Generally speaking whatever speed you see is simply the speed of the printer itself based on the settings you have selected, the software you are using, and the speed of your computer.
You can refer to the information on the HP website. It is just possible that replacing the printer driver MIGHT help but other than that there simply isn't much you can do about the speed of this (or any) printer unless you want to choose a lower quality print setting.
HP Photosmart C5250 All in One Printer Product Information