Generally speaking, it doesn't matter whether you use British English or American English in a story, so long as you keep it consistent:
- If you use "sceptical", you will need to ensure you use British English throughout ("colour", "car park", "realise", etc).
- If you use "skeptical", you will need to ensure you use American English throughout ("color", "parking lot", "realize", etc).
This can be difficult - as you've already discovered, it's not always easy to tell whether a certain word is spelt (spelled) differently between British and American English. Even I sometimes find myself getting mixed up between the British and American English spellings of certain words. However, most spell-checkers, including MS Word's and Google Chrome's, have separate settings for British and American English. If you use the wrong one by mistake, the spell-checker can flag it up for you.
There is one exception: in scientific writing, "aluminium" is preferred to "aluminum" (though both are correct), and "sulfur" is preferred to "sulphur". Unless you're writing a chemistry paper, though, you can most likely get away with using either version so long as it's consistent with the rest of the text, and only the most pedantic of readers will take issue with it.