The secret is put the circles in triangular formation and each group of 3 is the in alternate way upon and lower part against the walls of the rectangle. In the scheme that follows ABC represent the center of a group of 3 circles. The solution that Dario describes was the best known, published solution as of 5 years ago. Rotation and reflection are encouraged. Obviously, no proof, but I do have a diagram if you're interested.
All solutions use the repeated triangular pattern of circles with some adjustments at each end.
A reader noticed a posible improvement by adjusting the 5 circles on each end of the pattern and achieved 331 circles in a 2x165 rectangle. More recently, Nick Baxter improved it again by adjusting 7 circles at each end to achieve 329 circles in a 2x164 rectangle. Nick and I each examined things more completely, trying up to 35 adjusted circles on each end. We then tracked down a note by Hiroshi Yabe of Kyoto University who displayed the same result about 8-10 years ago.