What does a private detective do before and during a court in order to prove someone is guilty with his collected evidence?

If he’s working for the prosecution, he needs to give all the evidence to the prosecuting attorney.

If he’s working for the defense, he isn’t looking to accuse anyone; he’s looking for evidence to prove the accused didn’t do it, and he needs to give it to the attorney representing the client.

He isn’t working on his own. He is employed by someone, and he would give the evidence he collects to that person.
He then may have to testify as to what he found or saw, but that would be at the pleasure of the lawyer for the person for which he worked.

Detectives do not present evidence. They can testify as a witness for the prosecution.

I think you are getting your Perry Mason mixed up with your Ironside. Better back up a bit.

He has to convince the police department to arrest the suspect who in turn have to convince the prosecutor to take the suspect to trial. Then, no doubt, the detective would be asked questions to establish the case by the prosecutor and hostilely cross-examined by the client’s defense attorney.

Who hires the private detective? The suspect? The suspect’s family? The prosecution? The office of the district attorney? Has the grand jury met? Is there an indictment?

Generally speaking, private detectives do not work on murder investigations. This is an area of the law where the prosecutor and police force for the jurisdiction very jealously guard their monopoly on this activity. It’s not that it doesn’t happen, because it does occasionally, but it’s uncommon. Private investigators inserting themselves into an “open” murder investigation can be charged with obstruction of justice, and are often so threatened.

The police “in general” investigate the crime and collect the evidence that the prosecutor needs (or as much as he can have available, in any case) to make his legal case. After conviction, assuming there has been no hint of a conspiracy and no other criminals to indict for the crime (or for aiding and abetting, etc.), the case is considered closed and there is no more “official” investigation.

This is where you often see the movie treatment start, where the well-heeled convict, who still has friends and resources, hires the private eye to “prove his innocence”. If that case can be made, and it is a huge uphill fight when there’s already “a guilty man in jail, properly convicted”, then the convict’s lawyers will attempt to present their case showing an improper conviction, and to secure the release of their client. They don’t have a vested interest in making a conviction of someone else; it’s not their job.

The detective does not make the decision to make an accusation. The DA makes that decision based on whether or not there’s enough evidence.

So you have to work the plot so the family or someone integral hires him to find someone or prove something.