Hmm. The obvious route is the airport express train to Victoria then the tube (single line, no connections necessary) from Victoria to Euston.

The only ways you are going to do it without the underground are a taxi from Victoria to Euston, or to get the less frequent trains from LGW to the Midlands (they go to Bedford or Manchester) and then change somewhere along that line.

Try the German Railway site (yes really - it has very good coverage of the UK railways) to test out possibilities:

http://reiseauskunft.bahn.de/bin/query.exe/e