This is going to be tricky. If you stuck entirely to public transportation, you could take a bus from Lake Akan to Kushiro or Kitami, then take the Senmo line (JR Hokkaido) from Kushiro to Shiretoko-Shari station, then take a bus tour or something. As you can see, it is highly roundabout, and you might be better off taking taxi from Lake Akan to Mashu station, then take Senmo line up to Shiretoko-Shari station.
In any case, you have to be highly mindful of train/bus connection times because the service in those areas is usually highly infrequent.