If you go close enough to the speed of light, what you actually see is that space appears to shrink (in the direction of travel) and the trip seems to take less time than light would, because you've apparently covered less distance. Of course what those on the planets would see is that time has been moving oh so slowly on your otherwise speedy ship. There are equations you incorporate into a simulation that would account for this. If the game mechanics were such that you could could see what day/month/year it is in local time, vs your ships time, it would quickly become apparent that bashing through the void is no way to get anywhere.
Time dilation is just shortcut to say you are no longer sharing the same frame of reference. If everyone has their own frame of reference there is no difference between single player game and multi player one