Your relative has two things going on.
First, there is the drug addiction.
The drug addiction is an illness. This cannot be treated by kindness, hospitality, and Christian witness. He needs medical intervention.
Think about it–if your relative had cancer, simply allowing him to stay at your home and providing him with creature comforts (food, a bed, shelter from the elements, etc.) would not treat the cancer. Yes, there is such a thing as a supernatural healing from the Lord, and you and I have seen it happen (my husband was raised in a Pentecostal church and I was raised Baptist, and I’ve also seen this happen in the Catholic Church–I had an eye healed during a Catholic healing service), but these healings are rather rare.
Your relative needs medical treatment and rehabilitation for his addiction. If he doesn’t have health insurance, there are other options for him; e.g., the Salvation Army offers a program for drug addiction rehabilitation, and it is free. There are probably Catholic charities that help drug addicts to be healed from their addiction. Ask around.
Second, there is the homosexuality.
I mentioned in an earlier post that some drug addicts sell their bodies to anyone of either sex to get the money to buy drugs. If this is what is happening, then your relative isn’t a homosexual.
But if your relative has same-sex attraction, he is homosexual. This isn’t a sin, it’s a cross that God allows some people to bear because it is the best way for them to rely totally on God and become a saint (sinless).
I know this is different teaching than you have received in the Protestant churches. But it’s the truth.
Yes, acting on the homosexuality (having sex) is the sin, but the same-sex attraction itself is not the sin, and unless God intervenes in a supernatural way, your relative cannot be “freed” from same-sex attraction. I know that a lot of Protestant churches and organizations teach that a person with same-sex attraction should renounce it, marry a person of the opposite sex, and have children. This is not the answer for most people.
The answer is to pick up the cross and follow Christ.
One way that Christians with same-sex attraction can stay strong as they carry that cross is to join Catholic support groups like Courage. These people know how to help homosexuals because they deal with it in their own lives.
But FIRST, your relative needs to get treatment for his drug addiction so that he survives to bear his cross! Please ask around and keep searching for his sake, and find the appropriate (cost-wise) place for him to begin the treatment. That would be the kindest and most hospitable way to help your relative.