Legends of Tomorrow is a DC superhero show, yet another spinoff of The Dark Green Knight Arrow, and therefore has nothing at all to do with Doctor Who.
The Legends are a team of all the heroes and villains who were cooler than the leads in their respective shows: Firestorm, the Atom, Captain Cold, his friend Rory who dies but comes back, the sexy-dressing chick Rory wants who regenerated thanks to the Lazarus Pits, two new characters inserted just to add an interracial relationship, and, later, an older woman from a different time period. They're led by a renegade Time Agent, who stole a timeship to save history because his people weren't doing it, played by Arthur Darvill.
The team travels in a sentient timeship through a swirly circular tunnel called the time vortex, saving history. Their enemies include Reverse Flash, a supervillain speedster; Vandal Savage, an immortal viking; a guy who's a monk and likes to screw up history for the lulz; Damien Darhk, a superintelligent immortal who wants to fuck with the heroes' timelines so he can rewrite his own fate; and John Barrowman. Sometimes they also stop alien invasions of Earth, or a bad guy turns one of the team into a traitor to kill the leader, or they have to save a famous historical writer from enemies out of time, or the timeship comes to life as a hot chick and kisses the guy who stole her while two companions are in danger running through the ship's corridors. But the most dangerous threat is the leader of the Time Cops, who's actually a big asshole, and they need to find a way to stop him without letting the space-Nazi bad guys he's fighting win, which they eventually do by blowing up the Time Cops' home. In the next season, they search for the parts of a mystical artifact that can change reality that have been scattered through time.
The show avoids staleness by gradually rotating the timeship's crew. For example, after Rex Tyler showed up impossibly to give them a warning about the fate of the universe, they learned they should never go to Manhattan in the early 20th century, but they went there anyway, and now Hawkman and Hawkgirl are stuck in New York's past and they can never visit them again because of a fixed point in time. Shortly afterward, a mild-mannered nerd who always dreamed of being a superhero swallowed some alien tech and got superpowers, and he joined the team.
The series is accompanied by Supergirl, a spinoff for girls, Flash, a spinoff where people have sex with their own sisters, and Vixen, a spinoff nobody asked for starring a different version of one peripheral character.
They also have Pudding Brains, therefore it is one hundred per-cent canon to Doctor Who, making it the second-worst sci-fi show out there. This probably will result in the Beeb getting their grubby little fingers on this half-decent show, creating the worst crossover ever seen in the history of TV.