I assume you're talking about maintaining keys, in order, in an $O(N)$-sized array, with $O(\log^2 N)$ amortized worst-case update time.
At the beginning how many chunks do I have? I suppose one.
At the beginning, you will have one chunk. Ordered file maintenance doesn't make sense for very small arrays. Pick a smallish number (like 64 or 1024), and only start this whole chunking method once you grow above this small number. In the small array do something naive, pushing elements around as necessary.
When the array is full according to the target density, do I need to grow the array and how? I suppose yes, always by a power of two (number of chunks).
Yes, you can use standard dynamic array methods, like doubling in size when it gets too large.
Some data structures like to avoid this, as a full rebuild is a really large cost. But this isn't a big deal for us, as this data structure is highly amortized anyway. In other words, you have to do about $\Theta(\log N)$ rebuilds of the entire data structure every $N$ insertions anyway, just to maintain balance. This just adds an extra one.
Is there a possibility that a tree is impossible to rearrange according to the density? I hope not.
No, so long as there aren't too many elements. This is fairly easy to see---so long as you don't have too many elements, you can distribute the elements in the whole array evenly, meeting the density bound of the top array. But the arrays have looser requirements as they go down. So if the top array fits in density (and the elements are distributed evenly), the entire data structure is correctly balanced.
Is there any source code that I can take a loot at?
Yes, here (this was a class project; beyond that I know almost nothing about that code). The author of that code has a nice writeup of the data structure as well. Some more recent academic work on that data structure has experimental results. You may try looking through those as well (though of course, many are variations on the original concept).
When $N$ changes do I need to change the chunk size accordingly? I think so.
It depends on what you mean. Do you mean "when I insert a single element, so $N\gets N+1$, should I update the chunks?" then the answer is no. On the other hand, if you mean "when I run out of room in the data structure, and double its size, rebuilding the entire data structure, should I update the chunks?" then the answer is yes.