Blade of the Bloodchief's ability triggers regardless of who controlled the creature and whose graveyard it was put into.

The +1/+1 counters are put on the equipped creature, not Blade of the Bloodchief. If Blade of the Bloodchief is moved to a different creature, the +1/+1 counters will stay where they are.

The creature that gets the counters is the one that's equipped by Blade of the Bloodchief at the time the ability resolves. It doesn't matter what creature Blade of the Bloodchief was attached to (if any) when it triggered.

If Blade of the Bloodchief leaves the battlefield before its triggered ability resolves, the counters are put on the creature it was attached to at the time it left the battlefield. If it wasn't attached to a creature at that time, nothing gets the counters.

If the equipped creature itself is put into a graveyard, Blade of the Bloodchief's ability will trigger. However, it won't do anything when it resolves unless a spell or ability has caused it to become attached to another creature by that time. (Remember that you may activate an equip ability only as a sorcery.)

If the equipped creature and another creature are dealt lethal damage at the same time, the equipped creature will be put into a graveyard before it would receive any counters.