To end the standoff with the Illusive man via Charm / Intimidate, you need to have previously used Charm / Intimidate in every conversation with him.
There are 6 in total: Once on Mars, once on Thessia, once on the Cerberus Base, and three times in the final conversation on the Citadel.
The first three are hidden within left-side (investigate) branches of the dialog tree, and you can only use the final Charm / Intimidate if you have succeeded at all the previous 6.
The last paragon option will convince The Illusive Man to commit suicide, and the renegade interrupt will result in you shooting The Illusive Man before he shoots Anderson.
1: The Paragon/Renegade rewards for the Charm/Intimidate dialogue options on Thessia are switched around, so if you wish to maximize your Paragon points, you must choose Intimidate and vice versa. Doing so will not prevent you from convincing the Illusive Man at the end of the game.