I'm interested in finding out what the process of choosing the books of the New Testament was, when and who did the choosing
The canon developed gradually over the course of more than 300 years. In many cases, when decisions were made, they were simply to acknowledge what was already being read in the churches.
The process started early. Already in 2 Peter 3:16, there is a reference to the letters of Paul:
So at least some of the churches were already circulating Paul's letters and reading them as scripture.
The Gospels were written after Paul's letters, and the Apostolic Fathers quoted most often from Matthew, but also sometimes from Mark and Luke, and eventually from John.
By the late 2nd century, Irenaeus (Against Heresies 3.11.8) was claiming that the canon must contain exactly four gospels:
This was to counter fringe groups that were producing their own gospels, as well as Marcion, who argued for just one gospel (Luke).
We don't know who compiled the list known as the Muratorian Fragment, but it also dates from the second century and contains this canon:
Hebrews, James, 2 Peter, and 3 John were not yet accepted.
The fragment also recommends the Shepherd of Hermas as being worth reading but not qualifying for the canon because it was written "quite lately in our time".
By the early fourth century, the church historian Eusebius (Church History 3.25) sorts the known early writings into cateogories.
The first category is:
Next in importance are:
These are the writings that are universally accepted by the church. Eusebius adds that some believe Revelation should be included in this set.
The next category includes books that are disputed by some:
These would all eventually be added to the canon.
The next category includes books that rejected from the canon but worth reading:
He mentions that some would place Revelation in this group, and others would include the Gospel of the Hebrews.
And finally, Eusebius mentions other writings that are considered heretical, which he says should be "cast aside as absurd and impious." These include the gospels of Peter and Thomas, among others.
The first known list of canon that matches today's New Testament is found in the Easter Letter of Athanasius for the year 367:
Athanasius then adds that other books—the Wisdom of Solomon, the Shepherd, the Teaching of the Apostles, et al.—should be read by new converts. So even as the canon was being defined, there was recognition that among the excluded books were some that were part of the church teaching, and some that were not.
Let me give you the original document of Council Of Carthage AD 419
Here, the council take a decision to send the list to BONIFACE bishop of Rome, again in the council of Hippo, it was mentioned that "send the list to Rome and ratify it" It was the Roman Church who decided the canon, based on apostolic tradition..
This is hard to reference because it's a fairly organic process, but the works translated by St. Jerome into Latin to make up the Vulgate (People's) Bible are the ones that were read during early liturgies.
From a doctrinal stand point, this is the other answer:
When the Canon of the Scripture was closed, it already contained a list of books that were widely accepted as authoritative in the Christian communities spread all over the known world.
What we know today as deuterocanonical (or apocryphal) books of the New Testament were considered heretic very early in the history of the Christian communities. So there was never really a given time when people gathered and examined books to decide which ones would be included and which ones wouldn't.
Wikipedia has some interesting information about the Christian Canon.