It seems the opening of the letter to Philemon gives us the best probability of who Archippus was. “To Philemon our beloved fellow worker and Apphia our sister and Archippus our fellow soldier, and the church in your house:.”
J.B. Lightfoot ponders quite a bit about this person, first he concludes with fair certainty that he was the child of Philemon and Apphia, who had the slave Onesimus in their family:
With less confidence, but still with a reasonable degree of probability, we may infer that Archippus, who is likewise mentioned in the opening salutation, was a son of Philemon and Apphia. The inscriptions do not exhibit the name in any such frequency, either in Phrygia or in the surrounding districts, as to suggest that it was characteristic of these parts. Our Archippus held some important office in the Church; but what this was, we are not told. (SAINT PAUL’S EPISTLE TO THE COLOSSIANS AND TO PHILEMON, J. B. LIGHTFOOT, Pages 307–308)
As far as what ministry he had in the church there is less certainty but Lightfoot reasonable concludes that it was possibly head pastor (bishop) of Laodicea:
Laodicea was within walking distance of Colossæ. Archippus must have been in constant communication with his parents, who lived there; and it was therefore quite natural that, writing to the father and mother, St Paul should mention the son’s name also in the opening address, though he was not on the spot. An early tradition, if it be not a critical inference from the allusion in the Colossian letter, makes him bishop not of Colossæ, but of Laodicea. (SAINT PAUL’S EPISTLE TO THE COLOSSIANS AND TO PHILEMON, J. B. LIGHTFOOT, Pages 307–308)
This makes sense because “when this letter (Colossians) has been read among you, have it also read in the church of the Laodiceans…and say to Archippus, “See that you fulfill the ministry that you have received in the Lord.” This may have implied Paul was a little worried about potential lukewarm state of Archippus, which later could be part of the cause of that problem I the whole church later identified in revelation. (Rev 3:14-22)
So how sure are we that he was a church leader? ‘Quite sure’. How sure of what particular leadership role he had? ‘Not that sure.’
Note: Lightfoot mentions that there is no extra biblical information on the subject so we only have the scriptures to base our opinion on.