How does the play deal with masculinity?

One can answer your question partly by recognizing that, in the 60 years since Look Back in Anger premiered, our ideas on what actually constitutes masculinity have changed. Jimmy Porter seems driven to express himself as a man through aggression and abusiveness. Though he's an educated man, he prefers to make his living running a market stall. There is, of course, nothing wrong with this, but one senses that Jimmy chooses it chiefly in defiance of his wife Alison's middle-class background. Thus, to him, "masculinity" is somehow demonstrated by independence from or resentment of his wife's values. Jimmy also lords it over his friend Cliff, who functions as a good-natured sidekick to him. This is another way of Jimmy proving his "alpha male" status.
At the root of these factors, and of Jimmy's contempt for (and subsequent seduction of—another demonstration of his "masculinity") Alison's friend Helena, is anger. Jimmy Porter, of course, is the archetypal Angry Young Man of postwar theatre. Various elements converged in the 1950s to create this phenomenon: class conflict, post-WWII economic factors and disillusionment, and resentment against the changing status of women in society. Jimmy's assertion of masculine values seems, from our vantage point today, a kind of caricature. To some extent, Osborne probably intended it that way, because even from the perspective of his own time, Jimmy does not come off very well. Far from glorifying Jimmy, Osborne simply reports things as they were, and in looking back on Look Back in Anger today, we can understand that the dysfunctional dynamic of the Porter household, unfortunately, is not totally a thing of the past.

The characterization of Jimmy represents one of the most distinctly modern visions of masculinity. The "angry young man" is how Jimmy responds to life and how life responds to him. He is a man who is disenfranchised, the perpetual outsider to a world that is denying him a change. This is not the man of Homer's Hector, or Cervantes' Quixote, or even Joyce's Stephen Dedalus. Rather, Jimmy is the modern man who feels challenged at every turn. He recognizes that he is "better" than his job at the candy stall, and yet he cannot find his "foot in the door" for advancement. He recognizes what he sees as phoniness and inauthenticity around him and yet there is little he can do to change it. He believes that he could do better, while he is unable to really demonstrate anything to show this. He rails on about the need to view consciousness in a more "real" light, but he himself is unable to turn that high powered precision lens of criticism upon his own being. Jimmy is angry, and yet there is not a direct target of his anger.
In many respects, Jimmy represents the male who has been emasculated in different forms by the world around him. The vision of the triumphant, alpha male is not reinforced in this drama. Cliff and Colonel Redfern both represent men who are stunted by the world in which they live. The drama depicts masculinity in the modern setting as being poised between the past constructions of what was and a new world of what is. The result is men trying to find a path that is not illuminated for them. In large part, Jimmy's anger is directed at this vision of reality. Jimmy's struggle with the world in which he lives is reflective of what it means to be a man in the modern setting. It is a condition where the questions are many, the answers are few, and the only constant is insecurity amidst a world that constantly preaches that there is nothing about which to be insecure.