How do I kill the Jarl?

I always try to kill Jarl Balgruuf but he always comes back to beat me.

It's really annoying. It happens with most of the NPCs, not only those from Whiterun.

You can't normally. NPCs that are essential for the plot are immortal.

If you're on PC you can use the console to turn off Jarl Balgruuf's immortality, but on Xbox or PS3 you're out of luck. Note that killing essential NPCs can break your savegame, so do this at your own risk. If you still want to remove his essential status on PC, open the command console with ` (US keyboard) or § (some international keyboards) and enter the command:

setessential 00013BBD 0

Balgruuf will now be killable.

To kill other NPCs who have the temerity to annoy you with their immortality, you can use the same command, but replace the hexadecimal number with their BaseID instead of Balgruuf's BaseID. You can find an NPC's BaseID on their individual Unofficial Elder Scrolls Pages Wiki entries. Note that you need the BaseID, not the RefID, when reading those pages.