If you are staying in the west part of Kopavogur it is easy to access the main paths in Reykjavik. But other nice options are to run in the Kopavogur valley or around Kársnes.
The 12 kmroute goes through 2 of the nicest parts of the Greater-Reykjavik paths, Fossvogur valley and Kopavogur valley.
You can start at Kopavogur swimming pool (which means however that you have start on a hill and finish climbing 🙂
or start in Fossvogur which is easy to access from most parts of the city.
The 5.5 km route goes around Kársnes in the oldest part of Kopavogur.