I played Promising Future. My opponent played Relentless Pursuit to try and move into a battlefield. Does the showdown go through before I can do my 5 cards from Promising Future?

No, the showdown does not happen before you finish playing your cards from Promising Future.

According to the official rulings for Promising Future, all cards revealed and played during its resolution must be fully played and resolved before any showdown or combat begins.

Here is how the timing works:

1. Resolution of Promising Future: When you play Promising Future, it begins to resolve. Both players look at their cards, banish one, and then, starting with the next player, play their chosen cards one by one.
2. No Interruption: These cards are played as part of the resolution of Promising Future. Because a spell's resolution cannot be interrupted, your opponent cannot play Relentless Pursuit (which is an [Action] or [Reaction] card) to move a unit and start a showdown until Promising Future has finished resolving completely.
3. Showdown Staging: If the cards played via Promising Future trigger a showdown (for example, by moving a unit to a battlefield that becomes contested), that showdown is "staged." It will only actually start once the chain is empty and the turn returns to an Open State after Promising Future and all its associated card plays have completely finished resolving.

In summary, Promising Future must fully resolve first. Only after the chain is empty and the board state is finalized will any potential showdowns begin. Your opponent cannot interrupt your card plays to force a showdown before you finish.