He's particularly weak against champions that can keep their distance since he must relies on his passive to deal damage, and of course, the ult. Champions like Ashe will have an easy time since she can slow from a good distance. If he decides to try and get in too close or chase you, just kite him to your turret. Bring a flash if you need it to quickly get out of range if he grapples you.

Other than that, just don't give him kills. He's a selfish champion that relies on taking his team's much needed kills to make himself a lot more power. If he actually manages to get too many kills, simply target all of his teammates first and try to keep cc focused on Darius. Zilean's revive or a Guardian will waste Darius' ultimate, taking away most of his damage and removing the refresh.

A good Trynd can beat a bad Darius, but that's situational. I find that Jayce should theoretically Darius. He is ranged and melee, with a gap closer, a disengage, and long range poke with attack speed and movement speed buffs, he can deal damage while staying out of range of the grab + spin combo of Darius. Generally speaking, Darius OP.

try jax or irelia

Teemo. It's fun.

Kayle can counter him. Ranged harrass and an ulti that can put Darius' one on CD if you time yours right.

Nothing more adorable than a Darius that thinks he's the one with DoT . . . and then running Teemo against him.

Yorick, Singed, Cho'gath, Kayle.

Yorick absolutely wrecks Darius. It's almost embarrassing.

Also, Darius is a free kill for junglers. Gank top at level 2-4 (before he has wards), then keep coming back every time he steps toward the middle of the lane. With 0 escapes, no move speed buff and no CC, there's really not much he can do to escape except blow Flash/Ghost/Exhaust or turn and fight. If he turns and fights, it's 2v1, so you better be sure you can kill him 2v1.

I don't know how viable this actually is, but...

One time me and my friends did an all-support game for fun. I was Lulu (onhit) top against Darius, and I absolutely WRECKED him. I got to constantly harass him with my autoattacks (don't underestimate Pix's damage!), I could kite him forever with my Q, and if Darius ever actually pulled me in I could just W him and walk away unharmed.