Try going up to your tip toes, as if you were going to jump forward, but then put your heels back down. And melt your shoulders down and back as much as you can. Like some of the instructors say "put some up dog into your down dog"

1. bring your arms in slightly, towards each other
2. shrug your shoulders up and then let them slide down onto your back while engaging your core strongly
3. agreed with odonogc, try the tip toes trick as you do 1 and 2
4. you may even want to soften your arms slightly (NOT ALOT) to engage the shoulders to help take the rounding out of the upper back.

practice practice practice

looking good!

Chances are the heel movement is going to work and here's most likely why: your pelvis is tilted back slightly, which is causing the rounding in your back. The reason your pelvis is tilted back is that your hamstrings are tight - they attach to the bottom on the pelvis and to the shins, and are pulling the pelvis toward the shins. If you raise your heels, that gives frees up that portion of the hamstrings to tilt the pelvis, so the back can become straighter. Hope that makes sense. Most people cannot put their heels on the ground and have a long spine in down dog.

Here's some more:
1. try to move your sternum toward your hands
2. turn your hands out slightly, this can give some people mobility in their shoulders
3. move your feet further back - to me, it looks like you have a long torso, and you need room to fully extend it.

Your legs are nice and straight though - great job!

You might have kind of tight hamstrings. So I tell people to bend their knees a little bit and then sit the hips up on the ceiling. Move your weight back/press chest towards thighs/all that jazz. Then, as you loosen up your hamstrings, start lowering the heels to the mat (straightening the legs again) throughout the practice.

But your legs looked locked and straight. So lift the heels off the mat (sacrifice that satisfaction for a bit ) and bend the knees. Then you should be able to press the chest closer to the thighs and "sit" on the ceiling. Its ok to do down dog with bent knees! But, as your flexibility increases, you'll be able to straighten the legs and drop the heels closer to the mat.

Another sidenote: once you flatten out the upper body, be sure to pull the tummy in a tiny bit so you don't over-correct and have a swayback in your downdog.

edit: this is actually a GREAT picture of what Im talking about. You can see the difference of how bending the knees helps a bit ...
http://bodypositiveyoga.com/wp-content/uploads/2012/05/down-dog-spine-bent-knees-650x222.jpg