Boros Reckoner's first ability will trigger even if it is dealt lethal damage. For example, if it blocks a 7/7 creature, its ability will trigger and Boros Reckoner will deal 7 damage to the target.

If Boros Reckoner is dealt damage by multiple sources at once, such as by two creatures blocking it, its ability triggers once and one target is dealt that much damage.

Damage dealt by Boros Reckoner due to its first ability isn't combat damage, even if it was combat damage that caused that ability to trigger.