To be able to fight Henryk you have to follow Eileen’s quest line. To start her quest line, you have to talk to her before you defeat Gascoigne.
Once you’ve defeated Gascoigne, you have two options to continue her quest line. You can go the route I’ve chosen, which is to just buy the Hunter Chief Emblem from the Bath Messengers, or you could go and defeat the Blood-Starved Beast first. You will need to have opened all three gates within Cathedral Ward, before you enter the Forbidden Woods. As soon as all the gates are open (you can close the fourth gate which leads to the Monocular though) she should be standing just outside the Oedon Chapel, talk to her here once more and exhaust all her dialogue. If she isn’t standing there simply reload the area. If she’s still not there, go back and make sure that all the gates are open.
From this point forward the lamp at the Oedon Tomb, where you fought Gascoigne, will no longer be functional. Simply walk from the Cathedral Ward lamp back to Oedon Tomb. As you arrive, an orange-ish Hunter will be standing just a few steps away from below Viola’s corpse.
My strategy here is more of a passive one. I sneak up on him and start off with a visceral attack. After that I’m backing off and wait for Eileen to join the fight. While you can definitely try and fight him properly, I don’t recommend it as you can easily hit Eileen. Hit her often enough and she’ll start attacking you as well. Instead what I prefer to do here is to keep shooting at Henryk to interrupt him and let Eileen deal the damage. You should still be very careful about what you do here, as Henryk can easily kill Eileen. Also, make sure you keep a close eye on how many bullets you have left. I was quite lucky, because I royally screwed that one up!
Upon killing Henryk he’ll drop the Heir rune, now simply exhaust Eileen’s dialogue once again and be on your merry way… for now. It’s also worth noting that dispatching Henryk will unlock his attire at the Insight Bath Messengers within the Hunter’s Dream. This set has the highest accumulated resistance to Bolt in the entire game, so if you’re having trouble taking on bosses like the Abhorrent Beast or the two Darkbeasts for example, you might just wanna do a Henryk cosplay when you’re fighting them. While Eileen’s quest isn’t over yet, it won’t continue until after Rom, the vacuous spider has been defeated. As I mentioned earlier, the one important thing to keep in mind here is, that you have to do this part of the quest, before you enter the Forbidden Woods. If you enter the Forbidden Woods prior to dealing with Henryk, you’ve missed your chance until the next NG.