For me, understanding his passives were the key to building a team to overcome him.
Easy Pray Passive - Counterattacks when attacked by enemy that has no [Shield] buff.
You need a champion that can constantly keep up a shield buff to counter this. Shield sets only work for the first 3 turns, after that you need someone that throws up a shield on a short cool down. I use Lodric Falconheart, but you can easily get a rare like Valerie to do it. If you don't have this component, he's constantly counter-attacking which is extremely annoying.
Spectral Horror Passive - Decreases the damage received according to how much Borgoth’s MAX HP has been decreased. Damage received is decreased by:
90% if Borgoth’s Max HP has been decreased less than 10%.
60% if Borgoth’s Max HP has been decreased between 10% and 20%.
30% if Borgoth’s Max HP has been decreased between 20% and 30%.
0% if Borgoth’s Max HP has been decreased more than 30%.
Simply put, the more you decrease his MAX HP, the more damage he'll take. So you need damage dealing champs in Destroy sets, or enemy decrease max HP champs like Dark Elhain in order to decrease his MAX HP. You really want to get it decreased by at least 10% quickly because you do 1/3rd more damage to him after that threshold.
He's also vulnerable to turnmeter control, so take Armiger, and possibly coldheart. If one of them is in a destroy set they'll do very well.