2009-10 and 2010-11 we scored in 6 consecutive matches, and last season we scored in 7, I think
It's actually 8 last season surprisingly, though they weren't all good results:
February 13 Bournemouth 1 Stoke 3
February 27 Stoke 2 Aston Villa 1
March 2 Stoke 1 Newcastle 0
March 5 Chelsea 1 Stoke 1
March 12 Stoke 1 Southampton 2
March 19 Watford 1 Stoke 2
April 2 Stoke 2 Swansea 2
April 10 Liverpool 4 Stoke 1
Then we lost the next two games 4-0!
Edit: Though this is the longest run we've had in the Premier League of both scoring in every game and also remaining unbeaten