For reference, the oracle text: "At the beginning of each player's upkeep, destroy target nonartifact creature that player controls of their choice. It can't be regenerated."
There are a few ways you can make this ability asymmetric. One way is to have plenty of expendable creatures, whether they be tokens or otherwise. However, if you ever run out of creatures to kill, it can make it awkward to continue feeding. That said, black does have self-recurring creatures like
Reassembling Skeleton to provide fodder.
The other option I'll throw out is to just not run any creatures that die to it - creatures with shroud, indestructible, or protection can ignore it, as can artifact creatures. I could definitely see an
Erebos, God of the Dead creatureless control shell, or an artifact creature beatdown deck with
Armix, Filigree Thrasher. You could also include 'only conditionally a creature' tech like
Guardian Idol.
In the context of
Alesha, Who Smiles at Death.... it's a bit difficult to say what is the best way to make The Abyss good with Alesha. I guess one option is to have plenty of haste enablers like
Anger and
Lightning Greaves - if you can give Alesha haste, she can reanimate something to sacrifice to The Abyss. Lightning Greaves also pulls double duty here, by protecting a creature from The Abyss. Cards like
Legion's Landing // Adanto, the First Fort,
Bitterblossom, and
Kher Keep can also provide sac fodder if you want to lean into
Smokestack effects.