I have an experienced player wanting to create a tempest cleric who doesn't worship a god. His back story will be something along the lines of 'walking through a storm one day he is hit by lightning, which provides him with power'.
I'm happy to entertain the thought, but what aspects might go awry?
Best Answer
Yes. A cleric, druid, or paladin can choose to follow a cause they passionately believe in as opposed to a deity.
The PHB (see Appendix B) notes that it is perfectly acceptable for a DM to decide that there are no gods that are worshipped in his or her campaign:
Emphasis mine. In these instances where the players inhabit a faithless world, the clerics, druids, and paladins don't just disappear because there are no imaginary sky people to give them super powers. They draw their power from a devotion to whatever ideal they value above all others.
The DMG goes on to codify this a little more explicitly, and provides guidance on alternative religious systems to the standard polytheistic pantheon that the vanilla game assumes as the default. It even specifically discusses "forces and philosophies" as an alternative to deities entirely on page 13. It has a decent amount of text, so instead of quoting it in its entirety, I'll simply leave it to you to look up at your leisure. You might also consider animism, which the DMG discusses on page 12. Perhaps that storm was caused by an Air Spirit, and the act of walking through the storm without praying to it for safe passage angered the spirit, and now it calls the character to its service. There are many possibilities that are well-supported by the rules and do not require any substantial effort on your part to implement.