I got way too much time on my hands so since some people were curious about Gorillaz lore, here's a mostly comprehensive writeup of the band's story from beginning to present.
Inception: Murdoc was raised by an abusive dad and forced into showbiz, grew up to become a punk and tried multiple times to form a successful band to no avail. Lacking in instruments he attempts a heist by doing a high-speed ram into a music shop where Stuart Pot works at. After crashing into him, paralyzing and causing an eye fracture he's sentenced to 30,000 hours of community service tending to the vegetative Stuart and spends most of his time abusing him. While taking him on a joyride which ends with Stuart getting catapulted out of his seat during a drift he fractures his second eye and gets revived out of his vegetative state. Inspired by his "zombie looks", Murdoc has Stuart employed as keyboardist, vocalist and overall frontman of the band for being the ultimate pretty boy image for his project, and he's given the nickname 2-D in reference to the dents in his eyes. Murdoc travels to Soho to find a drummer, having heard rumors of a person possessed by the spirits of his dead hip hop friends, which has turned him into a musical virtuoso. Russel Hobbs is kidnapped by Murdoc and taken to Kong Studios, and while his first instinct is caving Murdoc in, he agrees to join the band after he realizes the material they have is really promising. 2-D's girlfriend is temporarily employed as the band's guitarist but after having an affair with Murdoc she's ousted by the band. Lacking a guitarist Murdoc places an ad in NME magazine which receives an almost immediate response in the form of a package, where a young Japanese girl comes bursting out shredding a solo with a Les Paul. The only English word she knows is "Noodle", the band employs her as their guitarist. Gorillaz is born.
Phase 1 / Celebrity Takedown: Gorillaz writes Ghost Train and Punk as their first songs and signs to EMI. Recording of their material took place in Kong Studios, a dilapidated mansion built on haunted ground that Murdoc got for cheap. Material is recorded and Gorillaz release the Tomorrow Comes Today EP and video as a test drive, gaining modest appeal in the process. Deciding it's time to go global and publicize marketing, Gorillaz record the Clint Eastwood video which turned them into animated superstars overnight. The Gorillaz album is released and G-Sides is shortly released afterward, mostly as an appeal to the Japanese market. After several music videos, live shows and even animated featurettes, Gorillaz split on bad terms after their disagreements on a script for a movie deal went south, and they all parted ways. Murdoc went to Mexico to party but was eventually arrested and sentenced to 30 years in jail. Noodle traveled to Japan to learn about her past and reconnected with her "mentor", where she learned that she was part of a covert government child soldier program trained in martial arts, linguistics and a unique trait for every child (in Noodle's case, music). She was originally smuggled to England after the program was decommissioned and evidence of it's existence was erased (which included euthanizing every child). Russel became a drifter and slowly lost his sanity after he began having visions of the grim reaper, who was hunting for his dead friends' souls. Del is exorcised at this point which sends him into depression. 2-D lives a lucrative and eventful life once he realizes that he's considered by many to be the "face" of Gorillaz which finally boosts his confidence, though this also turns him slightly egotistical and into a jackass.
Phase 2 / Slowboat to Hades: With a strengthened resolve, Noodle returns to Kong Studios. In the band's absence Kong Studios has been overrun by demonic forces and beings which takes her 6 months to clear out, after which she begins writing all the demos for what would eventually become Demon Days. Russel tries writing music on his own but none of it comes out right and much of it is haunted. Trying to get peace of mind he returns to England and discovers Noodle at Kong Studios, who he decides to help putting together Demon Days. Murdoc breaks out of prison after no sign of parole and cellmate murmurs that Parlophone were expecting a new album from Gorillaz. 2-D returns to Kong, realizing that Gorillaz wasn't something to just walk away from. Arriving alongside Murdoc, they soon settle into the same routine and work alongside material that Noodle and Russel had developed. Demon Days is recorded, largely inspired by Noodle's travels and having witnessed environmental deterioration at humanity's hands. Feel Good Inc becomes the band's triumphant return in visual and musical form. A bunch of advertising deals, rave reviews, music videos and tours later however, the shoot for El Mañana ends in a minor disaster after Murdoc's setpiece idea of blowing up the windmill (which he also did as an excuse to kill a psychotic fan-turned-executive who wanted to sabotage the band) ends with Noodle being MIA. Rumors arose that she was killed off or that she had gone on hiatus. At one point Murdoc had, in a drunken stupor, thought that she was dragged into hell by Kong Studios' monsters who were after Russel, and that Murdoc had gone on an adventure to save her, but this never happened. With Noodle nowhere to be found, Gorillaz fell apart and Kong Studios deteriorated even more. 2-D and Russel are grief strucken by the loss of Noodle, the former quits the band to get a law degree in Beirut while the latter continues fighting his demons.
Phase 3 / Escape from Plastic Beach: Things get really icky at this point. Murdoc burns down Kong Studios in an act of insurance fraud after it had been too contaminated and gets involved in shady business dealings selling fake weapons to pirates, amassing a fortune but also getting a price on his head in the process. Escaping to uncharted shores he discovers Point Nemo (eventually named Plastic Beach), an island that had formed by conglomerated trash building a landmass in the ocean. Murdoc crosses paths with the "Boogieman" (Sun Moon Stars), a gasmask wearing person who is described as a manifestation of everything bad in the world. Murdoc makes a series of deals with him to enable him making a new album, which includes the bringing of several collaborators from the world (and kidnapping 2-D). Murdoc never makes good on his end of the bargain however, and is soon not only hunted by pirates zeroing in on him but also the Boogieman. Plastic Beach is recorded under strained circumstances, 2-D being forced into performing, Murdoc constructing a cyborg version of Noodle to fill in her absence but also serving as a body guard, and using a drum machine in absence of Russel. At some point, an angered Russel goes insane diving off of a dock, swimming to Plastic Beach and eating irradiated food, causing him to grow in size. After missing for several years, Noodle shows up incognito on a cruise but is raided by pirates looking for Murdoc, and while shipwrecked she's saved by Russel, who both head to Plastic Beach. The band eventually reunites but it becomes short lived as Plastic Beach is raided by pirates. While Murdoc successfully evades the Boogieman who is also incapacitated by the Evangelist, the attack on Plastic Beach leaves the band quickly separated.
Phase 4 / We Are Still Humanz: Russel takes Noodle and escapes Plastic Beach but the former is mistaken for a whale and harpooned, leaving Noodle to disappear in the seas. Noodle is stranded and found by a fisherman, and while repaying her debt she accidentally unleashes a shape shifting hell demon that rises into the criminal underworld of Japan. After years of hunting it down and killing it, she evades it's minions by, nostalgically enough, packaging and shipping herself back to England and Murdoc. Murdoc who had escaped the pirates in a submarine with Cyborg Noodle was eventually tracked down by an EMI-owned battleship and then locked up in a secret Abbey Road "prison" before he was released to make new material for Gorillaz. Russel got washed up on a Korean beach where he was exhibited as a mythical North Korean kaiju equivalent, until the diet of regular food ended up shrinking him down back to regular size. He was released and later traveled back to England where he reconnected with Murdoc. 2-D had probably the worse end of the deal, as he got eaten by a great whale and had to live years inside it's stomach, eating whatever it consumed. Poor conditions lead to the whale quickly dying and washing up on shore, where 2-D spent months living marooned on the beach, before he realized that he was just off the coast of Mexico. Hitching a flight back to England, the rest of the band members are expecting him. Presumably at this point they temporarily stay in a flat that houses Andre 3000, the Boogieman and the Evangelist, and record DoYaThing prior to being evicted (though the canonicity of this is disputable since Russel is still large in the video).
At this point, Gorillaz presumably stays at the Spirit House while they record Humanz.