In "A Jury of Her Peers", what do we know about John Wright's personality? What kind of husband is he?

Any description of John Wright that appears in "A Jury of Her Peers" is made as indirect characterization. This means that the information comes from the other characters, or as a result of the actions of John Wright. The author will not directly state what he looks like, is like, or acts like. It all will be discovered by the reader, as the story progresses. 

This is a clever way of moving along with the plot. After all, the story is a suspenseful and consistent uncovering of facts and details that will ultimately lead to understanding what made Minnie Wright snap and kill John, her husband. 

All this being said, these are some traits we gather from John as the story goes:

1. He is petulant; has an attitude

....I don't think a place would be any the cheerfuller for John Wright's bein' in it.

These are words spoken by Mrs. Hale, the wife of one of John's colleagues, and also a former friend of Minnie Wright. It is Mrs. Hale who first begins to suspect that Minnie acted as a result of John's nonstop abuse. 

2. John Wright had the "feel" of a sociopath, that is, of someone who is unfit to lead a normal life unless he hurts or abuses someone. When the question is asked as to whether Mrs. Peters, the sheriff's wife, knew John Wright, her answer is that she has "heard" that he was, essentially, "good". Here comes Mrs. Hale again, though, and offers more insight into this belief:

Yes--good [...] He didn't drink, and kept his word as well as most, I guess, and paid his debts. But he was a hard man [...]Just to pass the time of day with him--.[...] Like a raw wind that gets to the bone." 

In an allusion that suggests foreshadowing, this is the very moment that Mrs. Hale notices the poignant, empty birdcage laying around. This is the birdcage where Minnie's only loving companion, a canary, was once kept. 

3. John was antisocial. When Hale goes to John Wright to ask whether he would get a telephone, particularly since they were so isolated from mainstream society, the latter put him off...

........saying folks talked too much anyway, and all he asked was peace and quiet--guess you know about how much he talked himself.

Moreover, John seems to also suffer from a degree of misogyny. John does not care about "folks", but he much less cares about his wife. The men knew as much. They didn't even bother going to Minnie first in aims that her female charm would work on her husband's disposition to get a telephone. 

 I said at the same time that I didn't know as what his wife wanted made much difference to John--"

Therefore, John Wright is quite a chauvinistic and unsociable character. We later find out that these traits also render him prone to abusing and bullying others. This is precisely what he does to his wife, ending in his demise.