Neila is the time of the chasimas ha'din. "Chasmeinu l'chaim...". Rashi in parshas Lech Lecha comments that the brachos given to Avraham, "v'e'escha l'goy gadol va'avarechicha..." etc refer to the fact that we mention the Avos in our shmoneh esrei -- Elokei Avraham, Elokei Yitzchak, v'Elokei Yaakov. Rashi continues, "yachol yehiyu **chosmin** b'kulan? Talmud lomar 'v'heyei bracha' -- becha chosmin." We end the bracha "magen Avraham" -- the chasima is only with Avraham's name. The Tiferes Shlomo quotes the Tikunei Zohar that Rosh haShana, "vaHashem pakad es Sarah...," is the holiday of Yitchak; the Tur (417) writes that Sukkos is the holiday of Yaakov, "ulmikneihu asah sukkos." Yom Kippur, says the Tif Shlomo, is the holiday of Avraham Avinu, and particularly neila, the chasima, is in his merit -- becha chosmin. Avraham personified the midah of chessed, and it is only through chasdei Hashem that we are given a chasima ltovah.
How do we earn chasdei Hashem? By emulating Avraham and doing chessed ourselves.
On erev Y"K there is a halacha of ritzuy. True, the shulchan aruch says this means that we should ask each other for forgiveness, but, as the Sefas Emes explains, that's not exactly all that ritzuy means. Tikabeil brachamim ub'**ratzon** tifilaseinu. Y'yehu na amareinu **l'ratzon**. We want Hashem to want to listen to us and want to accept our tefilos. How do we earn that? By doing ritzuy ourselves, which means that we should not just forgive, not just to tolerate, but that we should have "ratzon" for our friends -- we should want good for them.
Wishing you all a gmar chasima tovah, that our tefilos should be accepted bratzon, that we be zocheh to chasdei Hashem, and that we be zocheh to see the good in others and accept each other b'ratzon as well.