Singed and Twitch go bot. Gangplank goes mid. Teemo goes top.
Cassiopeia jungles, maxes out E.
When ganking bot, Cass uses the Triple-Trap-Wombo-Combo-of-Toxic-Terror: She, Singed, and Twitch all throw out their slows in a line behind the enemy. Singed tosses the target back after catching up while Twitch and Cass do facerolls on their keyboards.
When ganking mid, Cass lets Gangplank start with a Parley. Then, she places her slow behind the target and facerolls. Gangplank simultaneously places a Cannon Barrage right behind the enemy. Cassiopeia makes sure to stay between the enemy tower and the target so she can land her ult to synergize with Gangplank's.
When ganking top, Cass ganks from behind with an ult, allowing Teemo to place a mushroom on the target. They proceed to faceroll.
Aside from that, the teamfights basically follow your general meta strategy: ranged squishies stick behind the wall of meat that is the other teammates.
And there you have it, League toxicity at its utmost height. I have never done this before, and surprisingly, I have not seen it myself. I hope that does not change.
Anyone involved in a team that uses this build in ranked gameplay should be demoted to the bottom of Bronze 5 and not let out of the division.