Tychicus appears 5 times in the NT:
One of the brethren who accompanied Paul in his deputation for the offering for the Jerusalem church (Acts 20:4).
He served as the courier for Paul’s letter to Ephesus (Eph 6:21)
... as well as Paul’s letters to Philemon and the Colossians (Col 4:7)
Paul mentions Tychicus twice in his later letters, first sending him to Crete to be with Titus (Ti 3:12)
... and later mentioning to Timothy that he had sent Tychicus to Ephesus so that he was no longer with him (2 Tm 4:12).
Additionally, Baker's Encyclopedia of the Bible states:
Determining whether Tychicus accompanied Paul all the way to Jerusalem or whether he stayed in Miletus when Paul stopped there to greet the Ephesian elders is difficult. Though Acts 21:8 speaks of “Paul’s company” (kjv) as if all of the delegation remained with him, the fact that Tychicus is not mentioned with Trophimus in the Jews’ charge against Paul would seem to indicate that Tychicus was not in Jerusalem (v 29). Since he is often mentioned with Trophimus, Tychicus was likely also a native of Ephesus.
Most believe that he was also one of the two brethren (with Trophimus) who accompanied Titus in the delivery of 2 Corinthians (2 Cor 8:16–24).
Evidently Tychicus and Paul were close friends as well as fellow workers since Paul frequently refers to Tychicus as a “beloved brother.”
Eerdman's Dictionary also suggests he became a bishop in Chalcedon:
A Christian from Asia Minor, described as a beloved brother, faithful minister, and fellow servant (Col. 4:7; 2 Tim. 4:12; Titus 3:12) who served as a messenger and letter carrier for Paul (Col. 4:7, 16; Eph. 6:21–22). Tychicus (Gk. “fortunate”) accompanied Paul and delegates from the gentile churches in delivering gifts to the Jerusalem church (Acts 20:4). The name is attested in nine inscriptions from his native region of Magnesia. Late Greek tradition holds that Tychicus was one of the first bishops of Chalcedon.
Bibliography. S. L. Cox, “Tychicus: A Profile,” Biblical Illustrator 21 (1995): 79–80.
Per Holman Illustrated Bible Dictionary, he died a martyr's death:
Paul also sent Tychicus to Ephesus on one occasion (2 Tim. 4:12) and possibly to Crete on another (Titus 3:12). Tradition holds that he died a martyr.
In regards to Philemon, the Intervarsity Press Dictionary supports the position that Tychicu was Paul's trusted "faithful minister" and was actually the more important character in Paul's letter to Philemon.
This point is highlighted by the chiastic pattern of the text itself. Recall that a chiasmus is a literary device that arranges words and ideas into two parallel and inverted passages, with an odd member placed at the vertex, where the two passages intersect (ABCDC´B´A´). The odd phrase found at the vertex (D) helps the reader locate the passage’s principal idea. Consider verses 7–9 in this light:
A Tychicus will tell you all the news about me (v. 7a).
B He is a dear brother, a faithful minister and fellow servant in the Lord (v. 7b).
C I am sending him to you for the express purpose that you may know about our circumstances (v. 8a)
D and that he may encourage your hearts (v. 8b).
C´ He is coming with Onesimus (v. 9a),
B´ our faithful and dear brother, who is one of you (v. 9b).
A´ They will tell you everything that is happening here (9c).
The chiastic shape helps us to identify the most important ingredient in the instructions Paul sends to the Colossians: that [Tychicus] may encourage your hearts. Paul’s chief interest is that his ministry continue through Tychicus during his imprisonment (see Lohse 1971:171). The chiasmus also subordinates Onesimus to Tychicus, for it is the latter who is central to Paul’s plans and additionally is called faithful minister. The credential added to Onesimus, who is one of you (4:9), suggests that his task is to help Tychicus gain entry into this Colossian community.