Siman 53 in Orach Chaim is rather long, and deals at length with the qualifications and disqualifications of a shaliach tzibur. You may wish to read through the entire siman. Since it appears from your question that you are particularly interested in disqualifications relating to sins the person may have committed, the following statement of R. Moshe Isserles from seif 5 is probably the most relevant:
מי שעבר עבירה בשוגג כגון שהרג הנפש בשגגה וחזר בתשובה מותר להיות ש"ץ אבל אם עשה במזיד לא דמ"מ יצא עליו שם רע קודם התשובה
One who committed a sin unintentionally, e.g. he killed someone unintentionally, and he returned in repentance, he is permitted to be a shaliach tzibur. But if he did [the sin] purposely, he cannot [serve as shaliach tzibur] because bad reports have still been circulated about him before the repentance.
R. Yechiel Michel Epstein explains the status of "unintentional" and "purposely" as follows:
Aruch hashulchan O.C. 53:8
מלשון זה משמע דבמזיד אפילו פעם אחת ולא מהני התשובה לענין התמנותו לש"ץ ובשוגג מהני לפעם אחת ותימא דבשוגג אפי' בכמה פעמים למה יפסול דעל שוגג לא שייך שם רע כמובן וצ"ל דאין כונתו על פעם אחת דבפעם אחת אפילו במזיד כיון ששב מותר להיות ש"ץ דכיון ששב נתבטל השם רע מיד ופסולו במזיד אינו אלא כשעשה איזה פעמים ובשוגג גם בכה"ג כשר וכן משמע מדבריו בספרו דרכי משה ע"ש
From this language it is implied that [one who sins] purposely even one time [is disqualified] and repentance does not help for his being appointed to be a shaliach tzibur, and [one who sins] unintentionally [repentance] does help if [he only sinned] one time. And this is astounding, because in a case where [the sin] was unintentional, even if [he committed the sin] many times why should he be disqualified? Bad reports are not involved [if the sin was] unintentional, as is understood. We must [therefore] say that [R. Moshe Isserles's] intent was not to refer to a one-time [sin], for by a one-time [sin] even if done purposely he is allowed to be a shaliach tzibur because he repented, and since he repented the bad reports were immediately nullified.The disqualification for a purposeful [sin] is only when he has done it a bunch of times, and if [the sin was] unintentional then even in such a case [i.e. where he has sinned many times] he is valid [as a shaliach tzibur]. And this is implied from his words in his book Darchei Moshe, see there.
Later, R. Epstein discusses which sins R. Isserles was referring to:
ודע דזה שכתב הרג את הנפש לאו דוקא וה"ה עבירה אחרת ויש להסתפק בעבירות דרבנן כששב אם נקרא יצא עליו שם רע אם לאו ונראה דאין חילוק בין דאורייתא לדרבנן
And know that this that he wrote "he killed a soul" is not exact. It is the same if he committed another sin. And it is subject to doubt whether rabbinic sins once repented are considered to have created bad reports. And it appears that there is no difference between biblical and rabbinic [sins].
Some of the commentators debate whether a purposeful sinner who repented can serve as a shaliach tzibur during the rest of the year but not for public fasts, because the disqualification of "bad reports" might only pertain to a shaliach tzibur for public fasts.