Salary negotiation is a business conversation. The organization isn't concerned that your rent has gone up or your kids' school expenses have increased. Your co-workers are probably dealing with similar situations, anyway.
Negotiating salary isn't about personal value but professional value. Remember that each role in an organization has a limit for which they are willing to pay. Negotiate from a professional value perspective and keep emotions under control.
This conversation shouldn't be about getting your boss to take pity on you but to open their eyes to the valuable contributions you bring to the organization through your role.