[@Lucius Cypher]
Sure, I'll find the section in my DMG real quick.

Says here that it will be the "grappler's" Athletics/Acrobatics versus the target's Acrobatics to successfully climb on. While in the creature's space, you move with it and have advantage on attack rolls made against it (friendly reminder that advantages do not stack). You can move around as if on diffcult terrain, but be wary: the other creture can force you to a contested check of creature's athletics versus "grappler's" Athletics/Acrobatics and on a failure off you go.

And given you mentioned instant kills, the rule for that is if you ever end up in negatives at or or below the opposite number of your maximum hit points, you die. Falling deals 1d6 damage for every 10 feet you fall, to a maximum of 20d6. In short, you won't want to find yourself soaring through the air without wings.