Both can be accomplished be filing documents with the court. It is likely something that would be best done with the help of an attorney. Having the charge vacated requires a motion made to the court requesting a judge's order. Restoring your gun rights would be accomplished by filing a petition in much the same manner. The website below has examples of the documents I'm talking about.
The foregoing is commentary regarding a general legal question. It is not intended to be legal advice specific to the reader's individual situation. Answering your question is not legal advice and does not create an attorney-client relationship. You are encouraged to contact a qualified attorney to discuss your legal situation.
You cannot vacate this conviction because it is considered a crime against persons. However, you can get your right to possess firearms back by petitioning the same court that sentenced you or the superior court in the county where you currently reside. To be eligible, you must not have had any convictions (of any kind) within the past five years. Also, if you've ever been convicted of a class A felony (like first degree assault, for example), you are not eligible to restore your firearms rights. Make sure the felony assault on your wife was not first degree assault.
Although I do not believe you can vacate your assault as it is a crime against persons, you should be able to restore your right to possess firearms. As the other attorneys mentioned, you must petition the Superior Court to do that. An attorney can make the process easier for you, and help ensure that you meet the qualifications. That said, if you can muddle through the requirements and ensure that you meet them, then you could do it yourself. Some of the requirements are not written very clearly, and it can be difficult to make sure that you have met them all. You could likely call a few attorneys in your area and get a free consultation to see whether they think you are eligible and what they would charge.