Assess where your potential significant other is in his grieving process.
While it is likely a widower will grieve for his lost wife for the rest of his life in one way or another, it is important that he has had time to grieve the loss properly before beginning a new relationship. Not having had enough time to grieve can cause the relationship to fail because of unresolved issues.
Go slowly with your relationship and let her dictate the pace.
Don't force a serious relationship if she is not ready. Do not ask her to remove traces of her former husband in her house, such as pictures or keepsakes, as these may be important to her and her grieving process.
Do not try and compete with the dead wife.
She was special in his life and you will not ever take her place. Instead, carve out a new place for yourself while remaining respectful of the time she spent with her now dead husband.
Allow your new significant other to celebrate or grieve on milestones, such as the deceased's birthday or their anniversary.
Do not try to take this away or compete with it. Instead, let her handle it the way she would like to.
Stay respectful of people that will always connect him to his former wife, such as children and in-laws.
They will forever be tied to your new boyfriend through his dead wife, and this should be respected. Do not expect them to love you and be excited about your new relationship at first, either. Some people, especially children, may feel resentful that their parent is moving on, so it is important to respect their feelings and take things slowly.