The “court of public opinion” is not a court of law, but the opinion of the general public. Lawyers (and publicists) help their clients in court, but often also appear on television and give newspaper interviews to impress the “court of public opinion.”
 
The phrase “court of public opinion” is cited in a Connecticut newspaper from 1794, although many early 1800s citations are from Great Britain.
   
 
9 October 1794, Hartford (CT) Gazette, pg. 3:
From the CONNECTICUT COURANT.
To the Honorable the Court of Public Opinion in the State of Connecticut.
May it please your Honors,
MR. WILLIAM EDMOND, has appeared a voluntary delinguent at your bar, and pleaded “not guilty”* to the charge of being the author of certain writings signed Cassius.
(...)
*See Hartford Gazette, October 2, 1794.
 
14 September 1803, Green Mountain Patriot (VT), pg. 1:
The heinous crime, for which I am indicted in the High Court of Public opinion, consists “in just suggesting my opinion, that these Missionaries” from Connecticut, “are of very little service, owing, among other things, to their loose and irregular conduct, in many instances.”
   
Google Books
July 1807, The Edinburgh Review, pg. 346:
The cause of the anti-monopolists, meantime, can never want advocates, either in Parliament or out of it; and all parties act, and every part of the cause is tried, in the open court of public opinion,—a judge who can neither be bribed, nor deceived, nor daunted.
     
Google Books
The Magic of Wealth:
A novel, Volumes 1-2
By Thomas Skinner Surr
Philadelphia, PA: M. Carey
1815
Pg. 82:
...you will not deem it a question of indifference whether you have an able advocate, or none at all, in the court of Public Opinion.
   
Google Books
Rationale of judicial evidence : specially applied to English practice. From the manuscripts of J. Bentham.
By Jeremy Bentham
London: Hunt and Clarke
1827
Pg. 523:
Under the auspices of publicity, the original cause in the court of law, and the appeal to the court of public opinion, are going on at the same time.