It's not clear yet what is his purpose beyond the wall but we know some things and therefore we can speculate.
First and most important: the White Walkers were created centuries ago before the long night by the Children of the Forest as a weapon to fight back the first men as can be seen in the TV show.
Second: after the White Walkers were created both men and Children of the Forest had to fight them. They both made an alliance and were able to defeat the White Walkers (the Azor Ahai legend is from this period of time). After this, they built the wall and for some time the Children of the Forest even provided the Night Watch with Dragon Glass in the case of the Walkers came back.
From this, we can speculate that the magic that created the White Walkers went wrong and they turned against everyone in their way. They could be just a weapon that is out of control (I don't believe so) or they could be conscious and want revenge or even want to conquer Westeros.
Personally, I tend to believe that revenge would be more plausible since the way the Night King (the first Night Walker) was created. He was just a man, probably a soldier from the first men captured by the Children of the Forest.
As commented below the wall was built after the Night King was created. (The Night King is the reason why the wall was built)
Again, we still don't know his objective this is just speculation.