Whether to tell my father-in-law depends on your family's situation and decision. Buying a car with a loan is a private decision, and parents don't necessarily need to know. However, if you think he should know the decision, or you want his support and advice, then you can consider telling him.
When communicating with your father-in-law, you can explain how this decision was made, what is the loan amount and repayment plan, and why you think it is appropriate to buy this car. At the same time, you can listen to his opinions and suggestions, but the final decision is up to you two.
In a word, you'd better consider your own needs and decision-making process before telling your father-in-law, and make sure that you two have reached an agreement. If you think he should know the decision, you can tell him appropriately, but the final decision should be in your hands.