This really has not been a feasible route since the last major ice age when the land bridge existed (sorry, 10,000 years too late!). First, there are no roads/paths/trails to the Bering Straights from anywhere in Alaska that would be reachable by normal transportation modes including a bicycle. There is the Iditarod Sled dog trail but not really passable in the summer months without gigantic tundra bike tires (see National Geographic article a few years back on biking across AK roadless). The only feasible way to get near the 50 mile gap between the USA and Russia is to fly to Little Diomede Island in the Bering Sea. Your could potentially ride a bike on the icepack in the winter but might need to have diplomatic papers to allow such a thing. Pretty cold time of year to do so (be ready for -60F, 50 knot winds, and very hungry polar bears who have been known to track humans on the ice pack- yum!). As far as I know, few individuals have made the transit across the Bering Straights unless they are Native Alaska Eskimo who have relatives on the Siberia side. You may want to stick to the roads in AK in the summer months (great for touring) and consider flying to Vladivostok in Siberia for something a bit more realistic in terms of trans-border crossing. Note: There used to be a direct flight from Anchorage to Siberia in the summer years ago but this may no longer be in service. Try out of Seattle. Good luck!