Why was there a painting of Rue?

It's not a part of the training center. It's what Peeta did for his assessment. He's been well established in the books and films as being a very talented artist – decorating cakes, and himself.

Peeta used dyes from the assessment room to paint an image of Rue and it's what earned him his 12 along with Katniss'.

From the book:

Peeta seems to be struck with the same reluctance I'm experiencing. "Well, I—I did the camouflage thing, like you suggested, Katniss." He hesitates. "Not exactly camouflage. I mean, I used the dyes."

"To do what?" asks Portia.

[...]

"Actually, I painted a picture of Rue," Peeta says. "How she looked after Katniss had covered her in flowers."

That being said, the fact that it's still visible in the film is a departure from the books, where it was removed before Katniss entered the room.

I think of how ruffled the Gamemakers were when I entered the gym for my session. The smell of cleaners. The mat pulled over that spot in the center of the gym. Was it to conceal something they were unable to wash away? "You painted something, didn't you? A picture."

"Did you see it?" Peeta asks.

"No. But they'd made a real point of covering it up," I say.

Peeta was the one that painted Rue, just before Katniss was called for her assessment. The entire scene is much different in the book, and essentially glossed over, leaving it to the audience to figure it out.

In the movie, we only see the painting, and no reference to it before or after the assessment. Just Katniss' reaction to it. It reminded her of Rue's death, and essentially became the straw that broke the camels back. This caused her to want to threaten and piss off the game makers by hanging the Seneca Crane Effigy, to remind the game makers that they can die just as easily.

In the books, chapter 16 has Katniss and Peeta discussing the process of choosing allies, and planning for their assessments. They say that they don't know how they are going to kill "these people", their fellow tributes, having gotten to know them better. This reminds them of Rue.

"I don't want them as allies. Why did Haymitch want us to get to know them?" I say. "It'll make it so much harder than last time. Except for Rue maybe. But I guess I never really could've killed her, anyway. She was just too much like Prim."

Peeta looks up at me, his brow creased in thought. "Her death was the most despicable, wasn't it?"

Then Peeta gets called in for his Assessment. Katniss has to wait almost an hour before it's her turn. She knows something is different, but there is no painting.

When I go in, I smell the sharp odor of cleaner and notice that one of the mats has been dragged to the center of the room. The mood is very different from last year's, when the Gamemakers were half drunk and distractedly picking at tidbits from the banquet table. They whisper among themselves, looking somewhat annoyed. What did Peeta do? Something to upset them?

Since Katniss believes Peeta did something stupid, putting himself in danger again, she reacts, attempting to deflect the danger onto herself, again.

I feel a pang of worry. That isn't good. I don't want Peeta singling himself out as a target for the Gamemakers' anger. That's part of my job. To draw fire away from Peeta. But how did he upset them? Because I'd love to do just that and more. To break through the smug veneer of those who use their brains to find amusing ways to kill us. To make them realize that while we're vulnerable to the Capitol's cruelties, they are as well.

Cue Seneca Crane. Or a lovely tribute to him. Naturally.

Chapter 17 follows up with Katniss and Peeta letting each other and Haymitch and Effie know what happened during the assessments.

As we all gather for dinner, I notice Peeta's hands are faintly stained with a variety of colors, even though his hair is still damp from bathing. He must have done some form of camouflage after all.
[...]
Peeta seems to be struck with the same reluctance I'm experiencing. "Well, I - I did the camouflage thing, like you suggested, Katniss." He hesitates. "Not exactly camouflage. I mean, I used the dyes."
[...]
I think of how ruffled the Gamemakers were when I entered the gym for my session. The smell of cleaners. The mat pulled over that spot in the center of the gym. Was it to conceal something they were unable to wash away? "You painted something, didn't you? A picture." "Did you see it?" Peeta asks.

"No. But they'd made a real point of covering it up," I say.

"Well, that would be standard. They can't let one tribute know what another did," says Effie, unconcerned.
[...]
"Actually, I painted a picture of Rue," Peeta says. "How she looked after Katniss had covered her in flowers."

There's a long pause at the table while everyone absorbs this. "And what exactly were you trying to accomplish?" Haymitch asks in a very measured voice.

"I'm not sure. I just wanted to hold them accountable, if only for a moment," says Peeta. "For killing that little girl."

"This is dreadful." Effie sounds like she's about to cry. "That sort of thinking ... it's forbidden, Peeta. Absolutely. You'll only bring down more trouble on yourself and Katniss."

"I have to agree with Effie on this one," says Haymitch. Portia and Cinna remain silent, but their faces are very serious. Of course, they're right. But even though it worries me, I think what he did was amazing.

"I guess this is a bad time to mention I hung a dummy and painted Seneca Crane's name on it," I say. This has the desired effect. After a moment of disbelief, all the disapproval in the room hits me like a ton of bricks.

All of this cut or never included in the film. Pointedly, that Peeta was as willing to stick it to the Gamemakers or the Capitol as Katniss was at times. Like when he came up with the secret love the previous Hunger Games or pregnancy the following scene.

And it was a bonding moment for Katniss and Peeta we don't see either:

"You'd have thought we planned it," says Peeta, giving me just the hint of a smile.

"Didn't you?" asks Portia. Her fingers press her eyelids closed as if she's warding off a very bright light.

"No," I say, looking at Peeta with a new sense of appreciation. "Neither of us even knew what we were going to do before we went in."

The results of this, not seen in this movie, is that both Katniss and Peeta receive a score of 12, the highest. Haymitch says this is so Katniss and Peeta get targeted by the others. In the first movie, Katniss' antagonistic William Tell shooting of the apple moment in her 74th assessment resulted in a similar high score.