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Asterions blade wounds the archer mortally before a black flame leaps out and seals the deal for the poor soul.
I have no complaints if you are houseruling this but I I would be remiss if I didn't point out that by RAW the fire damage from Green Flame Blade (I simply reflavored it to black flames) targets
a different enemy than the target of the melee attack:
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Black Flame Blade
• Evocation cantrip
• Casting Time: 1 action
• Range: 5 feet
• Components: V, M (a weapon)
• Duration: Instantaneous
As part of the action used to cast this spell, you must make a melee attack with a weapon against one creature within the spell's range, otherwise the spell fails.
On a hit, the target suffers the attack's normal effects, and black fire leaps from the target to a different creature
of your choice that you can see within 5 feet of it. The second creature takes fire damage equal to your spellcasting ability modifier.
This spell's damage increases when you reach higher levels. At 5th level, the melee attack deals an extra 1d8 fire damage to the target, and the fire damage to the second creature increases to 1d8 +your spellcasting ability modifier. Both damage rolls increase by 1d8 at 11th level and 17th level.