Does Volibear channel a rune exhausted when playing a Mighty card or also when he makes a card Mighty?

No — Volibear only triggers when you play a unit that is already Mighty at the moment it's played.

Relentless Storm reads "When you play a Mighty unit, you may exhaust me to channel 1 rune exhausted." The Mighty check happens in the trigger condition at the time of play — the unit must have 5+ Might when it resolves and enters play. Making a card Mighty afterward (via a buff, a "When Played" self-buff like Sett, or any later effect) is too late; the trigger window has already passed. Volibear himself doesn't even make cards Mighty — he only channels runes.

A card that enters already Mighty — from a static/passive effect (e.g., Trifarian War Camp) or printed 5+ Might — does trigger him, and the rune does enter exhausted. (Rule 706; FAQ #4060/#1160.)