How you become Hortator of House Redoran is going to depend on whether you are a member of the House or not and, if so, how high you've managed to climb in the ranks. Dealing with Redoran can involve a bit of running around, but all of the councilors are in the Ald'ruhn Manor District.
If you are already Grandmaster of the House, simply talk to the surviving councilors and you're done.
If you are a member of the House, but haven't reached the stratospheric heights of the rank structure, you'll need to talk to the councilors and dispose of Bolvyn Venim.
If you are not a member of the house, you'll need to suck up to the councilors (because you've got a serious disposition problem) and dispose of Bolvyn Venim.
In any event, your first stop should be Athyn Sarethi at Sarethi Manor in Ald'ruhn. Venim is holding Sarethi's son as a hostage against Sarethi's good behavior. Sarethi wants you to rescue his son from Venim Manor.
This task is not as difficult as it may seem. You'll need to dispose of a few guards between where the son is being held and the exit (let them attack you before taking them out), but once the path is clear, all that's left is the walk back to Sarethi Manor. Once his son is safe, Sarethi will bring the other councilors on board (you'll still have to talk to them) and you can turn your attention to disposing of Venim.
With all of the other councilors in your pocket, you'll have to fight Venim because he will never agree to naming you Hortator. But don't just kill him or you're hozed. You will have to get him to challenge you to a duel in the Arena. This is why you need the other councilors' support BEFORE you go talk to him. Once you have that support, he will offer the duel rather than simply refuse to name you Hortator.
Once he has issued the challenge, go to the Arena Pit in Vivec where Venim will be waiting for you. It's a straight slugfest. He has no spells and no magic items. He does, however, have a full set of Ebony Armor, a Daedric Dai-Katana (60% skill) and 181 HP, so be prepared to have a serious hurt put on you if you decide to stand toe-to-toe with him. For my part, I just climb up on the railing at the top of the pit and pepper him with ranged spells and missiles until he's toasted through and through. You get to keep all of his goodies, but it's going to be awfully heavy, so make sure you can carry it all.
Once Venim is out of the way, go back to Ald'ruhn and talk to Sarethi and he will name you Hortator of House Redoran.