If I was in a car accident and the person that hit me has liability limits of $25,000 but I am seeking $300,000 for a claim, how would I receive compensation?

I am sorry to hear about your accident.  It can be very difficult to obtain any compensation above and beyond the insurance policy limits for the at fault driver's insurance company.  Your only real option is to pursue the driver personally for your damages which are beyond what their policy allows.  There is a significant difference between the $25,000 policy limit and the $300,000 you are seeking.  Pursing this type of claim generally requires the filing of a formal civil lawsuit.  You must evaluate whether the person who was at fault for the accident has assets and the ability to cover any judgment you might obtain against them.  Usually the answer is no.  In general, people with significant assets, the amount of assets you would need to cover damages this large, would be carrying much more than $25,000 in insurance coverage.  I realize this is not a good answer for you.  Good luck to you.