Ok. First, check out bbqcritic.com for good and bad presentations.
I'm more a pulled pork and ribs guy, but do ok in chicken and briskets. Different for each, but know that gristle and fat will get you killed on the judges table.
Also, judging a few will help you; pictures are worth thousands of words...
Ribs. Sample from each rack to choose what you want them to taste like and to ensure they're meeting your tenderness target. Choose those also that look as perfect as you can. When cutting them, slice them with an extra sharp knife and make those cuts as straight and perfect (no meat fraying) as possible. Arrange them similarly.
Pork. It's changed over time, and there are definitely regional differences. Up until about 5 years ago, slices in you pork box in my region would've pretty much had you killled. Now, slices are acceptable, but it better be the money muscle, and it better not be mushy. For that matter, it's almost expect to have at least 1 MM in the box. Slice carefully. Dont arrange the MM so that it looks like a turd. Then, pulled chunks about the size of your thumb, because to an extent they hold moisture better. Some with bark, some without. Arrange neatly, and fill the box. Tubes help too. Again, about the size of your thumb. Remove ALL fat!!! Do NOT have a fatty chunk in there.
Brisket. Separate the flat from the point carefully. Hopefully you'd taken as much fat off the the point as possible before cooking; it makes this next step better/easier. For the burnt ends, choose the sections with the least fat from the point and cut them into cubes; no fat! Make your ends as you prefer. Before slicing the point, we ALWAYS scrape all the fat from the bottom. Trim the slices to the exact same length (oh, slice with REALLY sharp knife, slices about the thickness of a pencil), and surround them with the burnt ends. IF the ends taste bad, or have fat in them, or aren't tender, DONT put them in there. Load in more slices if you have to.
On presentation, try not to have specs/flakes of anything on the meat that might take away from the presentation. Also, brush on some aujus (we mix in a TINY amount of sauce in ours) before closing the lid; adds heat and moisture. Or, pass the slices individually through a bath of this and then gently shake them off before placing in the box (dont want TOO much juice/sauce puddling).
Chicken. Frankly, stay with thighs. Bone in or bone out; all work. Breasts score well only if they're PERFECT, and that's PERFECT 30 minutes after you closed up the clam shell. It's nearly impossible not to thave them dry... Skin of course must be bite through. I think it's becoming more acceptable to present without skin, but some idiots (judges) get snooty if skin isn't there (not me, I dont like skin unless it's fried). Trim the chicken as much as reasonable as possible. Use muffin tins if you must, but the muffin-tin chicken thing has largely run it's course. When selecting chickens for presentation, try to get them very close to the same size. We sauce at the very end, and then they're dipped; no brush marks; no finger smears, etc. Place gently in the box, filling the box as much as possible, and as evenly as possible to increase eye-appeal.
We sample a chicken or two and make sure the skins are bite through. If not, REMOVE THEM and sauce them without skin. Rubbery skin is the kiss of death.