You're right, there is a conflict of interest. No matter how fair your family member claims to be, there will always be a doubt that you really received the contract based on your merit alone.
This can effect the way others in the company work with you, their attitude when dealing with you. They may not respect your abilities and may feel like you're getting a free ride.
As a result, the best way of dealing with the situation is for the family member to completely remove himself from the decision making process. He has no say. The decision is handed over to people who have no reason to be biased toward you and if you receive the job, it is based on your merit alone. Ideally, they should not even know you have a family member in the company. This way, if someone questions how you got the job, you can be 100% confident it was based on your merit.
If your family member can't remove themself from the decision making process, I would seriously reconsider taking the job or appling at all, since others in the company may not respect you or take your skills seriously.