A player the other night wanted to run, jump off a wall, flip over the other characters in front of him, land in front of the monster, and attack it — all in one turn. Is that actually doable?
Most of what you're describing is just flavor text. In game terms, your player wants to:
- Move adjacent to the monster (through another character's space)
- Attack the monster
Per PHB page 191, Moving Around Other Creatures:
You can move through a nonhostile creature's space.
Your player can dress it up however they like, but it's basically normal movement as long as all movement is through spaces that are either empty, or contain non-hostile creatures (though the occupied spaces count as difficult terrain).
If the movement moves them out of a hostile creature's reach as they go, they'll be subject to an opportunity attack.
The problem only occurs if they must pass through an opponent's square to reach their target square; AFAICT, the only PHB provided rules for doing so (following the quote above) are:
You can move through a hostile creature's space only if the creature is at least two sizes larger or smaller than you. Remember that another creature's space is difficult terrain for you.
Aside from that, per V2Blast's comment, the DMG (p. 272) offers optional Overrun and Tumble rules, that allow moving through similar sized hostile opponent's spaces based on the result of a contested Strength (Athletics) or Dexterity (Acrobatics) check respectively, and halflings have a special rule (Nimbleness) that allows moving through the space of a creature only one size larger than themselves (with no check required, just like moving through the space of an opponent two sizes larger or smaller). The DM might also allow you to try to jump a short enough opponent (though they'd be well within their rights to give the creature an opportunity attack if you did so) if your strength and movement speed were up to the challenge, but that would be purely a house rule.
To be clear, you must end your move in a free space (no doubling up players in a single square).