The way to write a letter depends on the topic that is being covered, but to write a letter to welcome or get closer to a daughter-in-law, a person should begin with a warm greeting, such as "My dearest (insert name)," and continue in a warm tone.
Ideas for writing could include memories of things that have been done together, memories of special times with the son and the hope to continue those special times with the daughter-in-law as well, and the sentiment that she is an important part of the family as a whole.
It is a great idea to include positive emotions you are experiencing with relation to her, such as excitement at her joining the family and the joy that comes with knowing that her marriage is making everyone happy. If you have not met or had much interaction with the future daughter-in-law, use the letter as a means to introduction, covering what the family enjoys doing together and welcoming her to enjoy any special annual family celebrations.
It is also a good idea to include some type of invitation to get to know her better, such as an invitation to the couple to come over for dinner or a potential future weekend visit. To close the letter, end with a warm sentiment, such as "Best Wishes," "Warm Regards," or "Love."