There are multiple ways they could do it.
For instance they could keep the "Schrader" name and say that he comes from a military family or was in the military himself (either way, based in Japan) and that he joined and was trained by the Foot Clan; particularly, Oroku Nagi, whom affectionately gives him the name "Saki". From there, you could do either of the 2 variations on The Shredder's story: Yoshi stealing Tang Shen from Nagi, Or, Yoshi stealing Tang Shen from Saki. Then say that he betrayed and killed Nagi, blamed his death on Yoshi (causing him to flee to NYC), took over the Foot Clan and then militarized it (replacing traditional ninja weaponry w/ military weapons; acting "dishonorably"), perverting what the Clan stood for in the eyes of Nagi, Yoshi and Splinter. Then of course, have him go on to murder Shen and Yosh, setting in motion the creation of the Turtles.
As far as his connection to Kraang, Basically do what they did in the 1987 series and just say that, through connections he made with TCRI,, he came into the employ of Kraang, who offered him an army (mutant and/or robotic foot) and continued weapon supply for his services (i.e. capturing the Turtles).