If my opponent has hissing miasma and I attack with enough to kill them, but doing so would kill me, would his enchantment kill me before my creatures even deal damage?
Hissing Miasma's ability triggers whenever a creature attacks, so during the declare attackers step (C.R. 508, especially C.R. 508.1m). If the life lost when that ability resolves brings a player to 0 or less life, that player will lose the game as a state-based action (C.R. 116.3b, 116.5, 704.5a), which happens even before any creatures that player controls can assign or deal combat damage (C.R. 506.1, 510).
If my opponent has hissing miasma and I attack with enough to kill them, but doing so would kill me, would his enchantment kill me before my creatures even deal damage?
Yes. You'll be dead before your opponent even has a chance to declare blockers. Combat consists of several steps, each one giving players priority, and that's when the game checks state based effects and says you're dead, as soon as you declare the attackers.
EDIT (May 18, 2019): Correct rule citations.
Yes. You'll be dead before your opponent even has a chance to declare blockers. Combat consists of several steps, each one giving players priority, and that's when the game checks state based effects and says you're dead, as soon as you declare the attackers.