How many NFL teams have started the season 0-5 and gone on to make the playoffs?

No team has made it starting at 0-5.
St Louis Rams, 2002 started 0-5 but didn't qualify:
http://www.pro-football-reference.com/years/2002

The question no one wants to ask must, indeed, be asked. Can this team -- a team that started 0-5 -- make the playoffs? Probably not. No team has ever opened the season with five losses and made it to January. Heck, only one team went 0-4 and made it. It'd take a special team and a unique set of circumstances to get it done. Probably more special than the Greatest Show on Turf, and more unique than the befuddling NFC. So again, probably not.
[bootlegsports.com/cgi-bin/bs3content.cgi?file=nfl-nflms-20021021164634] no longer exists

The team that started 0-4 and made it was the 1992 San Diego Chargers:

http://www.pro-football-reference.com/teams/sdg/1992.htm

No team in the 2003 play-offs started 0-5:
http://www.pro-football-reference.com/years/2003