Before Polaris's birth can be fully understood, it is necesary to determine what type of star it is. I have found two differing opinions as to what it is; one states that it is a binary star, while the other states that it is actually a collection of three stars, the primary of which is a Cepheid variable. Since these are not necesarily mutually exclusive, I will assume both to be true.
I have been unable, despite a great deal of searching, to find much information about the details of Cepheid variable formation, but what little I have found seems to indicate that a star becomes a Cepheid variable at the end of its life; the only prerequisites on the star are that it be roughly 3 to 9 solar masses. So the formation of Polaris was, most likely, much like the formation of any other star of just-over-solar mass. I will arbitrarily estimate that Polaris is a 4 solar mass star for the purpose of this project.
The evolution of 4 solar mass stars can occur in a variety of different ways, but all possible ways start off in a cloud of molecular dust. It is believed that solar-mass stars most often form in dark clouds, clouds with very low densities (10-15 mol atoms per m3) and very low temperatures (roughly 10 K). In any case, the cloud of dust slowly gravitationally collapse in on itself, with particles achieving greater and greater densities and temperatures. At some point, it will begin to glow in the infrared spectrum; strong sources of infrared light are often signs of these early "protostars". Eventually, the star will get hot enough for nuclear fusion to commence, and the star will start to radiate in the visible spectrum, much like any other star. At this point, this collection of matter can fairly be called a star.
But Polaris was not immediately a stable star. Before it could stabilize, it obviously had to stop collapsing. The specific amount of time that this took for Polaris is not available, simply because we were not there to watch it stabilize, but eventually the outward forces of the thermonuclear reaction at the center of the star did provide a precise counterbalance to gravity, bringing the star into the main sequence, the region around which it was, and is, to spend most of its life.