To find the low airfare, you can just search on google flights for sample dates on RT flights from LAX to 'Asia" (zoom the map to show Southeast Asia and HK), and then narrow down to which city (outside of China) has the lowest RT. Then, you look for any cheap options with connections in TYO. Once you have the dates and destination, you can play around with the multi-trip option, trying to build a stopover at the beginning or the end of the trip. You can add a filter like 'Star Alliance' so you will only get flights on *A airlines like UA and NH, which will help get rid of most of the super cheap Chinese airlines (which will connect in China instead of TYO).

To actually book, you just use the 'multi trip' option on any airline website or booking website. You can book: LAX-TYO; TYO-HKG; HKG-LAX (or vise versa). I have found that destinations all over Southeast Asia and Hong Kong all have much lower airfares, so you can try BKK/PNH/KUL/SIN/CGK in place of HKG. The trick is to find an itinerary that connects in TYO, which is why they are usually with Japanese airlines.