Emperor Mihail II lets you look at the top card of your library whenever you want (with one restriction—see below), even if you don't have priority. This action doesn't use the stack. Knowing what that card is becomes part of the information you have access to, just like you can look at the cards in your hand.

If the top card of your library changes while you're casting a spell, playing a land, or activating an ability, you can't look at the new top card until you finish doing so. This means that if you cast the top card of your library, you can't look at the next one until you're done paying for that spell.

You must follow the normal timing permissions and restrictions of the cards you play from your library.

The last ability triggers whenever you cast any Merfolk spell, not just when you cast a Merfolk spell from the top of your library. It will resolve before the spell that caused it to trigger, even if that spell has been countered or otherwise left the stack.