20 years after the 100 Year War, Toph Beifong had coitus with a currently unknown man (I kinda hope it's the Duke, but opinion may vary) and gives birth to the badass known as Lin MohterFuckin Beifong. Now Toph Beifong was very close to Suki and Sokka, so Toph made them Lin's godparents. Naturally the two graciously accepted.
So as the years went on in Republic City, both Suki and Toph became even closer. Both became enforcers of the law after Lin's father died (unless the father was Sokka, then someone very close to both of the ladies died instead) so the female duo decided to join the cops (and they would have made a badass cop duo).
So as Lin grew up, she was trained by her mother in Earthbending and Metalbending. But Lin was also taught how to fight without any bending by Suki. (who was, if I recall correctly, the first person in the Avatar universe to have Spider-man like powers [The Boiling Rock Part 2 when she climbed up the wall])
Meanwhile, Lin eventually starts dating Tenzin ( and we all know Sokka shipped them so fucking hard. I mean come on, his goddaughter and his nephew.)
Sadly, all is not well in the Avatar universe as eventually tragedy strikes and Suki gets killed while out on patrol. (hopefully not by a firebender, how unoriginal right? I kinda believe that an earthbender killed her) When Lin hears this, she is devastated. Before this Lin was like Toph was in A:tLA. But Suki's death made her realize that with great power (be it from great bending or not) begets great responsibility (I believe that Lin was a trouble maker, always getting into fights and it would be Suki telling her not to use her Earthbending for picking fights). So it is at this point that Lin decides to become a metalbender cop like Toph. She sacrificed a lot in order to become a person that Suki would be proud of.
Soo yeah. Sorry about any spelling or grammatical mistakes, my keyboard only seems to like every other letter. But to easily summerize that entire thing: