"by the back door" would mean that there are at least two entrances, one at the front and one at the back. Leaving through the back door is usually done to avoid press or anything unwanted.
You can use "through", but I would use 'by'.
From the point of view of those inside the courtroom, she can enter "from the back door"; but leaving "from the back door" is idiomatic only if the point of reference is outside the building. "She was able to avoid the press by leaving from the back door." This works (barely - by or through would be more common choices) because the press are outside the courtroom. To those inside, she left by or through, but never from, the back door.
The reverse is also true. To those paparazzi outside the courtroom, she enters the room by or through the back door, but not from it. In other words, in this context, "from" suggests that the person is approaching the reference point of the speaker/narrator. "She emerged from the front door to greet the waiting throng."