This is an awesome idea and I love your thinking. I have installed a similar system that has been in use for several years now. It works great - but you have a couple of hurdles to cross first.
First - you need pressure. Pressure to take your water input to the top of your tank (think: like a traditional water heater) this cold water is piped inside the tank depsiting the cold water near the bottom of the inside of the tank - again, like a traditional water heater does. I use a traditional water heater as my tank.
Typical water pressure inside a home is 40-60 lbs. This pressure you have provided will then allow you to fill your tub on the top floor. Electricity will be needed to provide the pressure in the system you have described.
Second - you will need a cold water return coming from the BOTTOM of your tank and into the water jacket at the stove. Here the water is heated and then it will thermosiphon up to an intake near the TOP of your tank. As the hot water enters the top of the tank, cold water at the bottom of the tank will circulate down to the stoves water jacket to be heated and sent back up.
Other things you should consider:
Pressure relief valves. One near the hot water exit from the water jacket and one near the top of the tank. THESE ARE NECESSARY!!
You have created a steam bomb. Extreme pressure will build in this closed system. If there is no relief and you burn enough fire without using enough of the water - It will blow up in a catastophic manner.
NO JOKE.
A second water tank. This will help in taking up the extra hot water created and put less stress on the system. Plus, it will be bonus to have extra hot water stored.
Once your system in in place and working - you cannot simply not use it any more. There must always be water in the jacket at the stove or it will burn up. Also - and most importantly - the water must always be circulating and being used or you will create the bomb that I mentioned earlier.
This link may be helpful in getting what your looking for:
http://inthewilderness.net/2017/01/27/homestead-hot-water-thermosiphon-loop/