Learn Finnish: Mihin aikaan elokuva alkaa? - What time does the movie start?

Mikä aika means "what time" in a general sense. When you want to say "at what time" an event happens, Finnish puts both words into the illative (the "into") case, creating the fixed phrase mihin aikaan.

Where does the word mihin come from?

It comes from mikä (what). In Finnish, question words have to match the case of the noun they are paired with. Since aikaan is in the "into" case, mikä also changes to its "into" form, which is mihin.

Why does the verb alkaa have a double "a" at the end?

The dictionary form of "to start" is alkaa. When conjugating it for a "he/she/it" subject like elokuva (the movie), Finnish often doubles the last vowel of the verb stem. The stem here is alka-, so adding another a makes it alkaa. It just happens to look exactly like the dictionary form!

Could I use the word kello (clock) to ask this question?

Yes, but you wouldn't use mihin kelloon. If you want to ask "at what time" using the clock, Finns usually use a different phrase: Moneltako elokuva alkaa?. Both mihin aikaan and moneltako are perfectly natural for asking when an event starts.