Play 'The Forsworn Conspiracy' like normal. If you have high enough speech you can persuade Eltrys to take a hands off approach, which will mean you don't get as much or any gold from him but he survives

When the guards try to arrest you, you can either fight them, intimidate them into leaving you alone, pull rank on them if you have finished the main quest or civil war, or go to Cidhna mine and do 'Noone escapes Cidhna mine'. if you persuade them they won't immediately try and arrest you but whenever you go to markarth they will try to arrest you as in vanilla. Alternatively you can avoid the guards and leave Markarth, which will finish 'The forsworn Conspiracy', and start the next quest. The guards will still try and arrest you next time they see you.

If you choose any of the escape options, 'The end of the Silver-Bloods' will start. You have the option of unilaterally killing them, persuading Igmund to let you kill them, or waiting until you have finished the civil war and getting your faction leader to help you get rid of them. If you want to unilaterally kill Thongvor, then you should put him into beedout then follow the 'Markarth will be better without you' path. If you do the first two be very sure you do NOT expect to be in a situation where Thongvor should become Jarl as finishing this quest will break that! You can still do this quest if you went to Cidhna mine, but if you kill Thongvor during the quest it will not be signposted until you speak to Igmund or the faction leader.

Once you have chosen your option, follow the quest markers. If you chose to do a civil war leader option, the Silver Bloods will attack you if you fail either of their speech options and you can kill them. After the quest, you can find them in Solitude jail, or their dead bodies in the wideadbodycleanup cell if you you asked Tullius to exile them. Igmund will become jaril again if he wasn't already. You should then either get a letter from Faleen or be forcegreeted by her next time you go near her (The latter only if you sided with the Imperials). She will ask you to go to Cidhna mine to ensure its security, and 'Violently loose ends' will start. If you unilaterally killed them, you can intimidate or persuade Igmund into cancelling your bounty or pay it to the guards as normal. If Thongvor was the Jarl., your bounty will automatically get cancelled. 

Go to Cidhna mine, where Urzoga will forcegreet you saying that the prisoners have escaped. You have the option to persuade her to help you look for the prisoners. You can look around the mine for a bit if you want, but the quest will not advance until you either leave the mine via the main entrance or enter Madnach's escape tunnel. If you do the latter, follow the tunnel as usual. Whichever you did, when you enter the the open air you will run into the Forsworn and they will will attack you- you're not a fellow victim of the Nords, you're an enemy agent acting on the behest of the Jarl's housecarl. You have no choice but to kill them and report back to Faleen for a reward. Don't forget to loot Madanach's body for the armour of the old gods!