There are a number of ways you can do the trip. Which is the best way? That is up to you. Do you want to go to ATH only?
A return award PER-ATH is 128,000 points + taxes and charges. One possible route is PER(QF71)-SIN(QF31)-LHR(BA640)-ATH(BA631)-LHR(QF32)-SIN(QF72)-PER. As I mentioned before you can route via HKG or even NRT.
If you want to visit other places, and have the time to do it, then why not redeem a OneWorld RTW award for 140,000 points (142,500 points if requiring phone assistance) + taxes & charges, including up to 5 stopovers and a limit of 35,000 flown miles. An example itinerary could be PER-SIN-LHR-ATH-MAD-JFK-LAX-HNL-SYD-PER.