Breakthroughs are found via anomalies and the omega telescope. You won't be getting to the omega telescope in time for this to be helpful so you will need to find the breakthroughs via anomalies. In my experience you can generally get ~3 breakthroughs through normal scanning and the rest are from deep scanning so you must put your research efforts into physics to get Deep Scanning. This must be your first priority. Once you have deep scanning finding the breakthrough is just a matter of scanning anomalies until you get the five.
You can get a leg up by choosing one of the commanders that start with a breakthrough, like the Inventor which starts with autonomous drone hubs. I can personally confirm this starting breakthrough counts toward the five.