Burden studied at the Maidstone College of Art and then took a degree in theatre design at the Wimbledon School of Arts. He took a job as a freelance assistant with the BBC. He worked mostly as a costume assistant on shows such as Filthy Rich and Catflap with Ed Bye.
Red Dwarf was the first show on which he worked as costume designer. He was asked to replace Jackie Pinks and completely revamp the costumes for Series III. He was responsible for several iconic costumes, such as Lister's leather jacket, The Cat's PVC suits and zebra print coat, Rimmer's shiny uniform and Kryten's bodywork. He later designed costumes for all kinds of life forms, the Canaries and even recreated Pinks' original Space Corps uniforms for Series VIII.
He has also worked extensively on films including The Young Americans, The Matchmaker, Tooth, The Hot Potato, and Crossmaglen. He has also worked on many television shows like The Bill, The Scarlet Pimpernel, Redcap, Whitechapel, and Robin Hood.
When Lister read a magazine during a visit to 21st century Earth, he saw an article about Chloë Annett, who played his missing love Kristine Kochanski. The article featured Annett with her two small children. These were actually the children of Howard Burden.