I just played Mordekaiser and since I'm smurfing I built him as a "tank" with Rylai's and FoN, with a Chain Mail in the middle for armor. This build I feel only worked because the other team got steamrolled by myself and our Ezreal. What is the real way to build him now, and what role should I fill in the team?

Not THAT different from pre-patch build.
Pre-patch looked something like
SV/Warden mail, Sorc shoes, Warden mail/SV, Frzn Mallet, Abyssal/Starks/Aegis/Warden into Randuins

now, it's something like

SV/Warden mail, Sorc shoes, Warden mail/SV, Rylai, Abyssal/Starks/Aegis/WotA/Warden into Randuins/Lichbane


As for role, a tanky mage, kinda like Vlad, but with a different style.
Try to sustain your presence and your damage output in team battles. When a foe gets focussed slap your Ult+Ignite combo on him and start focussing others down 1 by 1, constantly being in the fray as you try to keep your grey shield up.,

Sustained magic DPS, who can soak up some damage.

Magicpen, CDR, tankiness.

Sorc's shoes, Abyssal, Randuin's, Gunblade, AP, in whichever order needed.

I build morde as a AP/tank hybrid. Let's keep in mind that morde lacks utility as the teams only tank. Morde has no abilities that can keep your allies from taking damage. With that in mind, build survivability, and make sure to always go in first and be in the action to keep people off your allies.

I usually build:
Warmogs -> Merc Treads -> Rylais -> Thornmail(or FoN) -> Thornmail(Or FoN, whatever you didn't get last step) -> and for the last item, it's purely situational. Most likely an AP item, like Abyssal Scepter (Which also gives MR.)