The first thing you should do is have the cat checked by a vet. After he's neutered and gets his vaccines and is found to be in good health, he will need to be kept in a room where your cat cannot go. I used my bedroom, and was very careful going in and out!
A gradual introduction over a period of 12 to 14 days is best. I would use this as a guide:
1. No contact whatsoever for about 5 days. They will hear and smell each other, even though the door is closed.
2. Put a drop of vanilla behind each cat's neck. (They'll have similar odors, and that's important.
3. Prop the door open very slightly for several days.
4. Add an interactive toy that both can reach. Treats for being pleasant are in order!
5. When all is going well, put the new cat in a carrier, and put her in the same room with the other cat and you. Watch the interaction, of course, at each step. Treats for both cat will make it a pleasant experience. You want every step to be pleasant.
6. After 12 to 14 days total, release the new cat into the same room as your cat, and supervise! Treats and interactive toys again, of course.
Good luck. Try not to be too disappointed if the stray has feline leukemia. It's not unusual for a cat who has been running free to have the illness. A false positive is possible, so you might want to get another test in a month, if the result is positive.
I so hope all goes well. I'm sure you'll get some other hints. Let us know how things progress, please!
I kept my birds in their very well secured cages. I would not trust a new cat. Perhaps you'll get some ideas about that if and when you adopt the cat.