Absolom, a former pupil of Forest Hill School, made early appearances in the Sun-Pat Peanut butter advertisements before making his acting debut with the 1991 film Antonia and Jane which was followed by roles in several television series such as The Bill, Dangerfield and Silent Witness.

Absolom joined the BBC soap opera EastEnders in 1997 and had few storylines in his first year on the show. The storyline which gained him recognition was the Saskia Duncan murder storyline which began in February 1999. His character Matthew Rose was framed for the murder by the real culprit Steve Owen (played by Martin Kemp) Matthew's last appearance was aired in February 2000, and he won the Best Actor Award at The British Soap Awards that same year.

Since leaving EastEnders, Absolom played the leading role in the 2002 British horror film Long Time Dead and has played supporting roles in ITV television series such as Vincent. He is perhaps best known for his principal role in Doc Martin playing Bert Large's son Al. The part, since the series began, has won him critical acclaim. Absolom guest starred in The Bill between 1992 and 2009 in different roles, and as Benny, a violent loan shark in Casualty. He appeared as a Criminal in the internet crime thriller Girl Number Nine, also starring Tracy Ann Oberman and Gareth David Lloyd.

Absolom won a Celebrity version of TV show Total Wipeout which aired on 18 September 2010, receiving £10,000 for charity.