How one plays an unfamiliar song will vary from person to person. For some players, just saying, "play by ear" is sufficient. If that works for you, read no further.
For me, however, a little more explanation is helpful. It's easy for me to get lost in an unfamiliar song, and the role of the violin among singers is easy to get wrong. There are many ways to skin a cat, and if you don't need my explanation, that's all the better, but here's how I approached one song.
Below is the original:
What worked best for me was to just listen to the video and approximately dictate the melody in the Sibelius music notation program (I've gotten to the point where I can do it as fast as by hand). I then googled the chords to the song and typed or wrote them in.
In rehearsal, we played around with alternate harmonies and had a lot of fun doing so. As far as I've found, the most creative musicians in this respect have been those who perform by ear for the majority of the time, rather than by sheet music. The process of harmonization is for them an entirely intuitive process, although after they have come up with these chords, they are easily able to identify them. Although with experience it is not hard to harmonize a song intuitively, it is much harder for multiple players to agree on intuition alone. In our case, with piano, guitar and violin, we spent a considerable amount of time debating explicitly what chords to use. Below is the result of our process:
Additional liberties are possible in the introduction, bridge and outro. I shared the melodies at the beginning and end above with the pianist, with a few simple modifications as we saw fit.
Unfortunately we didn't record it, but we may make covers in the future. After rehearsing three times, I didn't need the sheet music, so I played by ear and probably gave the appearance of just 'playing along.' Hopefully in the future I will be able to do this more intuitively, but writing things out took me maybe only half an hour and was very straightforward.
In summary, I encourage any classically trained violinist or violist to step outside the box of published sheet music, play with musicians who can play by ear, and at least give the appearance of doing so themselves!