As Abukspan mentioned, the connection between Yom Kippur and Purim is already mentioned in Tikkunei Zohar. The Lubavitcher Rebbe explains the connection at length according to chassidus (originally explained by the Baal Hatanya in Torah Ohr).
The idea in short is that when we do an aveira we break the regular connection to Hashem which is dependent on our fulfilment of mitzvos. Teshuva means that we reach a higher level of connection which is not affected whatsoever by our aveiros. When we tune in to this connections, we are cleansed from our aveiros.
Like Yom Kippur, Purim is also associated with this connection, which is the reason that Hashem’s name isn’t mentioned in the Megilla, since it is connected to Atzmus – higher than any name or description. This is also the idea of עד דלא ידע on Purim – we reach a level which is completely higher than any intellect, even higher than the regular intellect of Torah kvayochol (בין ארור המן לברוך מרדכי).
See at length here: https://hebrewbooks.org/pdfpager.aspx?req=19850&st=&pgnum=42