You would need to prove:
1.) That what was stated, was FALSE. NOT a difference of opinion, NOT that he misunderstood something, NOT that he jumped to conclusions on incorrect or misleading evidence, but that he knowingly and deliberately LIED. His opinion, even if it disagrees with yours, is protected and NOT defamation. "You did a lousy job on the Peterson account" is an opinion and not actionable. "We have proof that you stole $20,000 from the payroll account" is a lie (unless they actually do have proof that you stole $20,000 from the payroll account).
2.) That he communicated, published, or otherwise caused these falsehoods to be made public to an audience. Anything he says solely to you is not defamation. Likewise anything he may put in your personnel file. Anyone who has access to your file is a "party at interest" who is entitled to know what he said in your discharge letter - that is not the same as making the word available to the general public.
3.) That as a result of the above communication, you suffered damages. Essentially, in this situation, you would need to prove that you failed to find other employment SOLELY because of this. In the current job market, that may be a tough standard to meet.
And yes, the burden of proof is entirely on you.