To satisfy the completion of this mission you are only required to be at the table when someone else is knocked out of the game.
Chuck can play with a friend during co-op mode. Use the magazines that give gambling bonuses to get more advantageous hands. This allows you to complete the mission in a shorter period of time, before the military arrives or other missions expire.
It is not required for Chuck to knock Jack out of the game in order to win the Knight Helmet. Any of the other players can knock him out of the game and the player will still receive the helmet, but it's advisable to at least try to beat him a few times before then.
Neglecting to rescue the other survivors tied to this mission (Kristin, Woodrow, Cora, Trixie-Lynn) will result in a one-on-one match against Jack. This method, along with playing on co-op, may be the simplest way to defeat Jack and obtain the Knight's Helmet, considering that when you have other survivors in the game, and thus more money on the table, Jack's skills usually cause his winnings to sky-rocket, making him increasingly difficult to knock out of the game.
If Chuck eliminates Jack, he will unlock the item "Arthur's Helmet", which is a part of the Knight Armor.
If Frank eliminates Jack, he will unlock the item "Protoman's Helmet", which is part of the Protoman Costume.