This sounds complicated and drawn out but shouldn’t take too long. You can do this at any point when you have control of the Normandy. Enter any of the system clusters using the Relay station, once you enter a cluster you can go to the edge of your current cluster and explore the rest of the system. You will notice a percentage next to the names of each cluster. Make sure you have enough fuel and make your way to each cluster.
Once you enter an unexplored cluster fly around the planets and enter their orbit by pressing . Back out and repeat this for each planet within each cluster. Once you have visited all the planets within a single system the achievement will unlock.