This song is named after the famous Western movie actor. The theme from Eastwood's movie The Good, The Bad, and the Ugly can be heard periodically throughout the song. Eastwood is most famously known for his portrayal of a western character with no name in Sergio Leone Italian westerns of the late '60s. He played Dirty Harry Callahan in the movies Sudden Impact and The Dead Pool, and was the Mayor of Carmel, California in the '80s. Eastwood is also a huge fan and supporter of jazz, and is a somewhat accomplished self-taught jazz pianist.
The reason the song is called Clint Eastwood is because of the strange harmonica-like instrument used in the background, which is called a melodica. It is a keyboard instrument that is played by blowing through a tube and pressing its keys. Damon Albarn of Gorillaz plays it throughout the track, and he thought it sounded a lot like the score to the Clint Eastwood's movie A Fistful of Dollars.
Also, Clint Eastwood is the name of a Jamaican Reggae artist who the band heard about when they were recording the album in Jamaica. Albarn explained during a Reddit AMA: "We were recording in Jamaica and listening to a lot of dancehall music and we imagined a cool moniker to have would be Clint Eastwood. Also I'm a great fan of the actor and of Sergio Leone and Ennio Morricone."
The "Laika Come Home" remix of this song is called "A Fistful of Peanuts." (thanks, Gina - Reston, VA, for above 2)
This garnered the band three MTV Video Music Award nominations, so they created an animated acceptance speech to continue the theme of cartoon band members. Unfortunately, they spent tens of thousands of dollars on a speech they never used, because they lost in all three categories.
The music video introduced most of the world to the four animated band members of Gorillaz (2-D, Murdoc, Noodle, Russel). The real mastermind behind the music is Blur frontman Damon Albarn. The characters were created by comic illustrator Jamie Hewlett.
Albarn sang the chorus. Del Tha Funkee Homosapien rapped the verses.
The lyrics, " I got sunshine in a bag" are probably a reference to marijuana, or at least to contraband of some sort. The line comes from the movie The Good, The Bad, and The Ugly when Eastwood refers to a saddlebag full of gold as sunshine in a bag.
Seven million global sales of the Gorillaz album earned the group an entry in The Guinness Book of Records as the world's most successful virtual band.