General assumptions: Another way to get to the holding cells is the smartest, because you'll take over the identity of defendant in a current court hearing. Choosing this method will allow you to leave the courthouse very quickly but you want able to explore the building closely.
The person, whose identity you can still is Timothy Hawke and you can find him in the courtroom (screen above). Of course I do not advise attacking him immediately, because it won't get any good. Instead of that, you have to somehow break the hearing, so you can eliminate Hawke in some secluded place and take his clothes.
Theoretically the trial can be stopped by disguising as the judge and considering Hawke innocent, but this is bad idea, because you'll lose the time unnecessary (once you get judge disguise, you can leave the courthouse anyway) and as the result of this decision, Hawke will stay in public places later on the mission. Incomparably better solution is to stop the video, which will be displayed as material evidence during the hearing.
First place, from which you can stop the video, is a balcony on courthouse floor, shown on the above screen. This place is accessible for everyone, so you do not have to sneak here.
A computer terminal to interact with is located left from the entrance to the balcony. This area is inaccessible for unauthorized persons, so you have to stay unnoticed. What's more, on higher difficulty levels, the computer terminal is guarded by an additional guard. So try to distract all policemen on the balcony. It is worth noting, that you can also reach this terminal from the nearby shutter (in order to get there, go to the courthouse back), but it shouldn't be necessary.
If you have safely got to the computer terminal, interact with it. The very important information is fact that the terminal becomes active only after few dozen seconds from the mission start, specifically at the moment, when both sides in the trial agree to watch the video evidence.
Do not worry, if you haven't managed to get to the terminal computer on the balcony, because there is another one in the area. In order to reach it, you have to get to the judge's chambers. Start with heading to the exit door, located on the ground floor. Once you leave the building, turn right, stopping after getting to the archive room shown on the above screen. I recommend stunning the guard inside, hiding his body and putting on court security guard disguise. This is of course not required, but will make easier your further actions.
Head to the parking lot, which is patrolled by many guards, even on lower difficulty levels. Start bypassing them, using vehicles and small walls (screen above) as covers. You can also try to distract enemies by throwing objects and turning on car alarms.
You can get inside judge's chambers through one of two shutters. I do not recommend the shutter shown on the screen 1, unless you play on lower difficulty level. It is much better to get to the shutter shown on the screen 2, because even on higher difficulty levels it shouldn't be observed. Regardless of your choice, you have to grab the ledge, climb up and finally get inside the building.
Explore a small right room in judge's chambers. A computer terminal you're looking for is shown on the above screen. Also this one will become active in a particular moment during the trial.
Regardless of whether computer terminal you have used, you'll stop the trail. Timothy Hawke will go to the toilet and this is the best place to attack him. If you hurry up, then you can get there (men's room) before him. Otherwise go to the women's room and use the ventilation shaft shown on the above screen.
You can wait for Timothy inside the shaft or in the closet located in the toilet. Leave your hideout in the proper moment, surprise your victim from behind and stun or kill him. Now you can put on Tin Foil Hat Man disguise.
Exit the toilet and return to the courtroom accompanied by a guard. Place yourself in the spot shown on the above screen and wait for the judgment.
Now you can head to the door leading to the holding cells. It is located on the right side (screen above). No one will stop you in the meantime, but make sure that you do not accidentally enter the evidence room - you do not have an access to be there.