We ashkenaz start selichot on a Sunday. Sunday is a day of the week. Usually we start events on a given day of a given month (e.g., rosh hodesh, rosh hashanna), or after a given number of days (e.g., we count 49 days to shavuot). Why, and by what logic, does the selichot custom differ from this usual approach to specifying a date?
this article cites a reason given by Leket Yosher Orach Chaim, p. 118 par. 1
Simce people have more free time on SHabbat, they spend their time learning Torah and enjoying (Oneg) Shabbat. So,Yom Rishon is a good time to begin Selichot since people are still happy from Shabbat. And we also learn in Talmud Shabbat p 32b that the shechina (G-d's "presence") exists not because of laziness or sadness, but rather because of the joy in fulfilling a mitzvah.