-Kryten (Pete Part I)
Nothing is revealed of Baxter's origins in the television series, except that Baxter at some point in history committed crimes severe enough to get sentenced to Adelphi 12.
In transit to Adelphi 12, Baxter was one of 400 of the Galaxy's most hardened criminals held aboard The Tank on the restricted Floor 13 of Red Dwarf, where Baxter quickly gained a reputation as the most unpredictable and violent inmate, and sycophants rallied round him. Baxter would steal the food and possessions of others, and relished in beating people up, even his own cronies on occasion. Baxter was also the resident prison bootlegger of illegal moonshine and GELF Hooch.
Baxter was killed by the Cadmium II radiation leak that wiped out the crew, but resurrected by the nanobots three million years later. Baxter soon took up his old habits and also again his old position as the most feared inmate of The Tank.
Baxter was an extremely large and aggressive felon, with a very short temper and a casual disregard for the safety of others and even his own safety. Rimmer insisted that Grizzly Bears ran screaming from Baxter, and offered as proof of his courage the fact that he had once bet one of his own testicles on a pair of Jacks during a poker game. (Only the Good ...)
A deleted scene from the episode "Pete II" suggests that Baxter was a virgin, and part of the reason he was so mean was because Baxter was sexually frustrated (the same explanation is given with Kill Crazy).
When The Cat bumped against Baxter in the prison canteen lunch queue and spilled his soup, Baxter shoved The Cat's head into the food dispenser and ordered hot Bovril. He then stole The Cat's lunch tray and stormed off.
In revenge, Kryten used the Time Wand against Baxter, turning the roast chicken on his tray into a live chicken. Baxter's neighbour found this very funny, until Baxter elbowed him in the face. He then looked around for the source of the anomaly and saw The Cat laughing at him. As he leapt Towards The Cat, Kryten froze him in mid-air along with the rest of the crew (RD: Pete I).
Later, Baxter was part of the Canary unit sent down to the food decks in the shuttle train to eliminate the Tyrannosaurus Rex rampaging there. Baxter and Kill Crazy insisted on tackling the dinosaur hand-to-hand (a T-Rex was both Baxter and Kill Crazy's idea of a fantasy brawl) and tried to attack Rimmer and Lister to stop them using the time wand. Their punches had no effect, because the wand had placed Rimmer and Lister's bodies out of sync. They settled for stealing the time wand, which did them little good as they couldn't work it properly, de-evolving themselves into ape creatures. Lister was able to restore them to normal before he destroyed the wand (RD: Pete II).
When Kryten needed to play a devious prank on Lister, he stole Baxter's stash of illegal alcohol and hid it in Rimmer and Lister's cell. When they realised their cell was about to be searched, Rimmer and Lister tried to drink what was left. The hooch, however, was almost pure alcohol and got them very drunk almost instantly. When he learned what had happened to his hooch, Baxter threatened to do something very unpleasant to Rimmer and Lister, prompting them and the rest of the gang to escape The Tank and the ship. (Only the Good ...)
Baxter was presumably left to die aboard Red Dwarf along with the other convicts as the ship was eaten away by the Chameleonic Microbe, and as the rest of the crew escaped on the shuttles in the Series VIII finale.
However, a deleted scene for the episode would suggest that Baxter survived, as Baxter gained a place on one of the shuttles - Lister had actually won this lottery, but Lister was intimidated into offering his place to Baxter instead.
Behind the scenes
Originally, Baxter was to have been turned into a giant baby by the time wand, which involved Ricky Grover wearing a romper suit and a bonnet. This scene can be found as a deleted scene on the Series VIII DVD for "RD: Pete I".
There was also a scene written for "RD: Only the Good..." where Lister is offered a place on the escaping shuttles, and is about to offer his place to Kochanski, but gives the place to Baxter in return for his life.
Exactly what crime Baxter had committed to get into The Tank was never mentioned, but it is safe to assume it was nothing pleasant.