This Guide will spoiler you!
You should read them after you played the "Story Mode" except you don't care.

There are 3 achievements that you can only get in the "Challenge Mode".

They are:

Call me "Tails"
Finish Challenge Mode in 2 minutes.

Don't Go Astray
Finish Challenge Mode in less then 500m.

Iron Fox Mode
Finish Challenge Mode, but only die once.

To get through the fox's den later, you have to chatch three animals.
Luckily there are some right on the way, but you have to hurry.

The counters only start when you cross the starting line.

After two easy jumps you come to a tree. Crawl under the tree and you'll see the first butterfly.
Normally you could also catch the butterfly with the spider web, but we don't have the time.
Instead, you can jump from the root to your left onto the branch under the spider web. From there jump onto the butterfly.
This won't always work because the butterfly flies in circles.

Then you have to jump over two pits.
After the pits, you have the choice of runnung straight ahead or going down.
Run straight ahead and jump on the stones in the pond.
Soon after you come to the second butterfly and a fallen tree.
You can make it a little easier to catch the butterfly by jumping right in front of the tree.
Crawl through the tree and then jump on it. From there, with a bit of luck, you can catch the butterfly.

You have to crawl through the tree again.
Keep running and jump over a hedgehog. Soon you'll reach your den.
In front of the fox's den there is a mouse hole and tall grass.
The mouse only leaves its hole when you hide in the tall grass.
The regular way is to wait until the mouse is far enough away from you and then chase it.
But you don't have to be completely in the tall grass for the mouse to come out.
If you're only halfway in the tall grass, the mouse will come out.
It can even come to you and let you eat it voluntarily.

The movements of the mouse are random.
If you go for the Don't Go Astray achievement, you can stay there and wait till the mouse come to you.
But if you go for the Call me "Tails" achievement and the mouse does not go directly to you. You have to chase it regularly.

Now that you have captured three animals, let your partner guide you through the den.
This is where the really hard part begins.
After crawling through a second fallen tree, you are pursued by hounds.
There is not much to say here, except try to stay calm so you don't make any mistakes.
Only one place exist here where nothing will happen to you. It's the top of the roots from the third fallen tree.

At the end, you'll get shot.
This counts as one death, so you have to complete the game perfectly if you want to get the Iron Fox Mode achievement.
Good Luck!