In the original movie the snow fell on the day before Christmas but it is not specified how many days it took for the children to build him. Is it assumed he was built within a single day on Christmas day?
On December 24, between 3:00pm and 4:00pm frosty came to life for the first time. By dusk, frosty came to life for the second time, and began counting and speaking.
My source, the film:
In the opening narration, the narrator, clearly states that what follows happens on the day before Christmas.
The children get out of school at 3pm on the dot, and they go straight to building a snowman, shortly after they place the hat on Frosty and he comes to life and says "happy birthday". Moments later, the hat is removed. A short time later the hat returns and is placed on Frosty's head again.
All of this happens before the sun sets, and appears to take place before dusk, so I think it is safe to say definitively that Frosty's birthday was December 24th.
What remains unclear, is the year of his birth. I don't think there was any mention in the film to a specific year.
Frosty The Snowman entered popular culture as early as the 1950s - source: https://en.m.wikipedia.org/wiki/Frosty_the_Snowman
An extensive breakdown of the films, songs and publications depicting Frosty's adventures can be found here http://toonopedia.com/frosty.htm