Kylo and Hux were already in a relationship before the events of tfa, they only faked to dislike each other to hide it from Snoke. But he discovered it anyway and used it to manipulate Kylo. He told him that if he didn't kill his father (or anyone else who was close to him before he joined the First Order) if he met him again, then Hux would pay for it with his life.
That's why Kylo looked so conflicted before killing Han Solo, a part of him really wanted to go back to the good side, but he knew he had to choose between his father and his lover and in the end he couldn't bring himself to give up on Hux.
Killing his own father left him broken and hurt, even though he tried not to let anyone see how he really felt in order to keep his position in the First Order, so he could keep on protecting Hux.
+Hux didn't know about it and he only discovered it later and then tried to comfort Kylo as best as he could.
+Hux becomes very protective of Kylo and does everything to protect him from Snoke's schemes.
+ Kylo and Hux relationship was actually a healthy and loving one, not really fitting with the standards of the First Order.
+Maybe at one point they consider leaving the First Order, but they have no idea where to go since everyone in the Resistance must hate them now because of the death of Han. (It's up to you if they finally decide to leave or to stay)
+Kylo and Hux somehow knew each other before joining the First Order.
Kylo Ren / Hux - Snoke blackmailing Kylo Ren, Hurt-Comfort
That's why Kylo looked so conflicted before killing Han Solo, a part of him really wanted to go back to the good side, but he knew he had to choose between his father and his lover and in the end he couldn't bring himself to give up on Hux.
Killing his own father left him broken and hurt, even though he tried not to let anyone see how he really felt in order to keep his position in the First Order, so he could keep on protecting Hux.
+Hux didn't know about it and he only discovered it later and then tried to comfort Kylo as best as he could.
+Hux becomes very protective of Kylo and does everything to protect him from Snoke's schemes.
+ Kylo and Hux relationship was actually a healthy and loving one, not really fitting with the standards of the First Order.
+Maybe at one point they consider leaving the First Order, but they have no idea where to go since everyone in the Resistance must hate them now because of the death of Han. (It's up to you if they finally decide to leave or to stay)
+Kylo and Hux somehow knew each other before joining the First Order.