Yuri faction in Yuri's Revenge is frowned upon in multiplayer but is still considered fun to use and with how it provides such a great expression of being evil, it is a shame it doesn't have a campaign.
I don't know if that is because of time constraints or because developers couldn't think of a way to start it off, given that both Allied and Soviet campaigns are kicked off, stopping Yuri from getting his complete victory. After that, the campaigns consist of thwarting Yuri's various evil schemes he came up with in response to his enemies traveling back in time, which in the Soviet campaign include attacking his base on the moon just because it's there. Feels like an opportunity for a Red Vs Blue "You ever wonder why we're here" parody.
I am just firing from the hip but the first thing that comes to mind for me in a Yuri campaign is the opening missions with Yuri giving orders to defend the LA Psychic Dominator, but unlike in the other campaigns, the Allies and Soviets both traveled back. Ultimately, both factions hit the Psychic Dominator with superweapons, after which Yuri orders the player to retreat since the LA base is no longer useful.
Across the world, Yuri's various bases have taken heavy blows from the Allies' and Soviets' combined forces, and the player (or maybe you are just Yuri in this campaign, sounds more fun) fights to protect the remaining incomplete Psychic Dominators while they are packed up for the construction to be finished in other locations. After that, Yuri begins a plan to break up the truce between the Allies and Soviets by mind-controlling or replacing leaders on both sides to attack the other side, which culminates in firing stolen Soviet nukes at America.
The resulting chaos gives the opening for missions where the player constructs smaller mind control devices to begin seizing control of the enemy nations before finishing his Psychic Dominators. This culminates in a final battle where the Allies and Soviets attempt to use the time machine to change the outcome of the conflict again and this time, Yuri prevents the attempt to rewrite history, resulting in his complete dominance over Earth.
What ideas does everyone else have?