How did the threefold distinction of Deacon, Priest and Bishop come about?

The first century writing known as the Didache only mentions apostles, or missionaries. However, Paul's use of the Greek term for deacon ('servant') indicates that this office dates from the early years of Christianity. Commentators continue to debate whether the role of priest also arose during the apostolic era. Certainly, the Shepherd of Hermas talks of 'elders' -- priests -- presiding over each of the churches early in the second century, but makes no mention of bishops.

Bishops are mentioned in the Epistles to Timothy and Titus, but almost all New Testament scholars believe that these epistles were written pseudepigraphically early in the second century. The word ἐπισκόποις (overseer) is also found in Philippians 1:1, but many scholars believe that the word was not used in the sense of 'bishop' in this epistle. Philippians 1:1 also mentions Timothy, who is closely associated with the role of bishop in the two epistles that bear his name, so the verse could even have been redacted in later centuries.

Francis A. Sullivan SJ has closely examined a range of first and second century Christian writings and, from these, argues in From Apostles to Bishops that the role of bishop did not arise until the second century. He says (ibid, page 15):

There exists a broad consensus among scholars, including most Catholic ones, that such churches as those of Alexandria, Philippi, Corinth and Rome most probably continued to be led for some time by a college of presbyters, and that only during the course of the second century did the threefold structure become generally the rule, with a bishop, assisted by presbyters, presiding over each local church.

By the end of the second century, Christian tradition held that the apostles themselves had appointed the first bishops by laying on of hands, and that bishops could only be appointed by existing bishops laying on hands, so that there is a continuous succession all the way back to the apostles themselves.

The early overlap in the three offices was primarily between bishop (ἐπίσκοπος - episkopos) and presbyter (or "priest", "elder"; πρεσβύτερος - presbyteros). "In the Apostolic writings," writes Russian Orthodox Protopresbyter Michael Pomazanski, "the two names of 'bishop' and 'presbyter' are not always distinguished."

Thus, according to the book of Acts the Apostle Paul called to himself in Miletus the “presbyters of the Church” from Ephesus (Acts 20: 17), and instructing them he said, Take heed therefore unto yourselves, and to all the flock, over which the Holy Spirit hath made you bishops (overseers), to feed the Church of God, which He hath purchased with His own blood (Acts 20: 28). However, from these and similar expressions one cannot conclude that in the age of the Apostles the two ranks — bishop and presbyter — were joined into one. This shows only that in the first century church terminology was not yet as standardized as it became later, and the word “bishop” was used in two meanings: sometimes in the special meaning of the highest hierarchical degree, and sometimes in the usual and general meaning of “overseer,” in accordance with the Greek usage of that time.

Orthodox Dogmatic Theology (3rd ed.), p.254

I don't think there was ever a time when there was overlap between the office of deacon (διάκονος - diakonos) and the two other offices. As Protopresbyter Michael describes,

Deacons, seven in number, were chosen by the community of Jerusalem and ordained by the Apostles, as we read in the sixth chapter of the book of Acts. Their first assignment was to help the Apostles in a practical, secondary activity: they were entrusted to “serve tables” — to give out food, and be concerned for the widows. These seven men were later called deacons, although in the sixth chapter of Acts this name is not yet used.

From the pastoral epistles it is apparent that the deacons were appointed by bishops (I Tim. 3: 8– 13). According to the book of Acts, for the ministry of deacon there were chosen people “filled with the Holy Spirit and wisdom” (Acts 6: 3). They took part in preaching, as did St. Stephen, who sealed his preaching of Christ with his martyr’s blood; and like St. Philip, who performed the Baptism of the eunuch (Acts 8: 5 and 38). In the Epistle to the Philippians, the Apostle Paul sends greetings to “the bishops and deacons” (1: 1), as bearers of the Grace-given hierarchical ministry, helpers of the bishops.

Ibid., p.255

By the time of Justin Martyr (100-165), deacons were also distributing the Eucharist, to those present as well as to those who were unable to attend the Liturgy: "Those who are called by us deacons give to each of those present to partake of the bread and wine mixed with water over which the thanksgiving was pronounced, and to those who are absent they carry away a portion" (First Apology, Chapter LXV).

Already by the end of the first century, the three offices were distinct, as witnessed by Ignatius of Antioch (35-108):

Be ye subject to the bishop as to the Lord, for he watches for your souls, as one that shall give account to God [Hebrews 13:17]. Wherefore also, ye appear to me to live not after the manner of men, but according to Jesus Christ, who died for us, in order that, by believing in His death, ye may by baptism be made partakers of His resurrection. It is therefore necessary, whatsoever things ye do, to do nothing without the bishop. And be ye subject also to the presbytery, as to the apostles of Jesus Christ, who is our hope, in whom, if we live, we shall be found in Him. It behoves you also, in every way, to please the deacons, who are [ministers] of the mysteries of Christ Jesus; for they are not ministers of meat and drink, but servants of the Church of God. They are bound, therefore, to avoid all grounds of accusation [against them], as they would a burning fire. Let them, then, prove themselves to be such.

Epistle to the Trallians, Chapter II

See that ye all follow the bishop, even as Jesus Christ does the Father, and the presbytery as ye would the apostles; and reverence the deacons, as being the institution of God

Epistle to the Smyrnaens, Chapter VIII

The practice of ordination of presbyters is described in the Acts of the Apostles and in the Epistles of Paul:

Acts 14:23 (KJV 1900)

And when they had ordained them elders [presbyteroi] in every church, and had prayed with fasting, they commended them to the Lord, on whom they believed.

1 Timothy 4:14

Neglect not the gift that is in thee, which was given thee by prophecy, with the laying on of the hands of the presbytery.

2 Timothy 1:6

Wherefore I put thee in remembrance that thou stir up the gift of God, which is in thee by the putting on of my hands.

Thus, concludes Protpresbyter Michael,

The Apostles, by the authority of Christ, established three hierarchical degrees, and that for the raising up of selected persons into these degrees there was established ordination, which communicates to them the active Grace of God which is indispensable for their ministry. It goes without saying that the successors of the Apostles, the bishops, had to fulfill precisely what had been decreed by the Apostles: that is, ordination through laying on of hands, joining to it the same exalted meaning and the same significance that were given by the Apostles.

Ibid., p.301

The Apostolic Canons provide the more formal guidelines that the Church adhered to:

Canon I

Let a bishop be ordained by two or three bishops.

Canon II

Let a presbyter, deacon, and the rest of the clergy, be ordained by one bishop

(Note: I am writing from an Orthodox perspective, but we share a common tradition with the Roman Catholic Church with respect to the early Church)