Bismihi ta'ala

If you are asking whether the second marriage contract will be legitimate, then the answer is yes, it will be legitimate, but if you are asking whether this is the right thing to do or not, clearly it is going to be the wrong thing. 

Let's say this man's first marriage isn't going as planned and there are problems, then he has a responsibility to fix these problems and try to amend his relationship. 

If his wife is giving him hardship, and nobody is able to solve their issues, then if they are beyond any salvation as far as the marriage is concerned, they should part ways and divorce. This would be better than living toxically together, probably leading to neglect, or even abuse. 

If the husband has no major issues, and just wants to be with another woman, he must remember his moral obligations and try to control his animalistic lustful desires. As Muslims we are taught to discipline ourselves, and not pursue worldly lusts without any consideration of anything. 

The wife also has feelings and needs, and if he does wrong her in these ways he will be oppressing her, and we know that punishment for oppression is very severe. 

With prayers for your success.