Is it DIYable? Yes. No. Maybe?
You may find that you start the job, everything works out just right, and you complete the work in record time and under budget. More likely, you'll run into a bunch of unforeseen problems, take 5 times as long as you'd expect, and end up way over budget.
The best way to tackle DIY work that you are not sure about, is to have the resources available to quickly fix mistakes. In your situation, it might be a good idea to purchase a replacement window before you begin. If you damage the old window beyond repair, you'll simply have to install the new one. On the other hand, if the repair works out, you can return the new window you purchased.
It also never hurts to have somebody with knowledge in the type of work you're doing, so if you run into trouble they can offer guidance.
If you come across something you are really not comfortable with, there are always professionals available to do the work for you. You might also be surprised that they charge less than you might think (not always the case, but does happen).
Since it sounds like it might be difficult for you to get to a lumberyard to build your own solution, you might be better off using a pre-made product. AC-Safe, makes Universal Air Conditioner Supports that might be useful in your situation.
The system requires you to screw the platform into the window sill (which may or may not be possible in your situation), then simply rests against the outside of the building to support the A\C unit.
I've also seen brackets that mount to the bottom of A\C unit itself, and offer support against the outside of the building (but I was unable to find an example online. I think Thermwell makes them).
Best Answer
My solution has been to build up the inner windowsill by temporarily affixing a 2x4 on top of it, and use that raised surface to mount the AC. Depending on the window hardware, you may be able to align this so the window can be closed normally when the AC isn't present, avoiding the need to dismount and remount the adaptation.
This could be made more attractive, of course. But an AC in the window usually means aesthetics aren't a primary concern.