The Rambam in Hilchos Deos Perek 6, Halacha 6 says:
6 When one person wrongs another, the latter should not remain silent
and despise him as [II Samuel 13:22] states concerning the wicked:
"And Avshalom did not speak to Amnon neither good, nor bad for
Avshalom hated Amnon."
Rather, he is commanded to make the matter known and ask him: "Why did
you do this to me?", "Why did you wrong me regarding that matter?" as
[Leviticus 19:17] states: "You shall surely admonish your colleague."
If, afterwards, [the person who committed the wrong] asks [his
colleague] to forgive him, he must do so. A person should not be cruel
when forgiving [as implied by Genesis 20:17]: "And Abraham prayed to
God..."
ו כשיחטא איש לאיש לא ישטמנו וישתוק כמו שנאמר ברשעים ולא דבר אבשלום את
אמנון מאומה למרע ועד טוב כי שנא אבשלום את אמנון אלא מצוה עליו להודיעו
ולומר לו למה עשית לי כך וכך ולמה חטאת לי בדבר פלוני שנאמר הוכח תוכיח
את עמיתך ואם חזר ובקש ממנו למחול לו צריך למחול ולא יהא המוחל אכזרי
שנאמר ויתפלל אברהם אל האלהים:
Based on this Rambam one should rebuke his/her friend that has done something to him/her that bothers them. The person should not keep it in, but rather share it with the other person.
With regard to your question of how to rebuke them, the Rambam continues and says:
7 It is a mitzvah for a person who sees that his fellow Jew has sinned
or is following an improper path [to attempt] to correct his behavior
and to inform him that he is causing himself a loss by his evil deeds
as [Leviticus 19:17] states: "You shall surely admonish your
colleague."
A person who rebukes a colleague - whether because of a [wrong committed] against him or because of a matter between his colleague
and God - should rebuke him privately. He should speak to him
patiently and gently, informing him that he is only making these
statements for his colleague's own welfare, to allow him to merit the
life of the world to come.
If he accepts [the rebuke], it is good; if not, he should rebuke him a
second and third time. Indeed, one is obligated to rebuke a colleague
who does wrong until the latter strikes him and tells him: "I will not
listen."
Whoever has the possibility of rebuking [sinners] and fails to do so
is considered responsible for that sin, for he had the opportunity to
rebuke the [sinners].
ז הרואה חבירו שחטא או שהלך בדרך לא טובה מצוה להחזירו למוטב ולהודיעו
שהוא חוטא על עצמו במעשיו הרעים שנאמר הוכח תוכיח את עמיתך המוכיח את
חבירו בין בדברים שבינו לבינו בין בדברים שבינו לבין המקום צריך
להוכיחו בינו לבין עצמו וידבר לו בנחת ובלשון רכה ויודיעו שאינו אומר לו
אלא לטובתו להביאו לחיי העולם הבא אם קיבל ממנו מוטב ואם לאו יוכיחנו
פעם שניה ושלישית וכן תמיד חייב אדם להוכיחו עד שיכהו החוטא ויאמר לו
איני שומע וכל שאפשר בידו למחות ואינו מוחה הוא נתפש בעון אלו כיון שאפשר
לו למחות בהם:
Fundamentally, the one giving rebuke is trying to convey that they are doing it for the benefit of the other person. People are perceptive and will pick up if you are doing it sincerely for their benefit or not.
Ones very tone and word choice should convey this attitude.
I hope this answers some of what you are asking.