So i see in alot of peoples runs that they are able to save replays of their time trial runs and also challenge their ghost times. How do you save replays of runs? im playing on xbox one and i got the game around release so i dont have the DLC characters is that why? is that feature only available if you have the DLC or the newer version of the game with Mighty and Ray?

Included in the Plus DLC.
While not required for run submissions as submitting just the replay alone is allowed, it's good practice to record your runs as they happen live since sometimes you might either forget to save the replay after a run (speaking from personal experience) or if a run makes use of pause buffering there's no way to tell from the replay alone what strats are being used.