Well the term "WOOD STOVE" and clean do not work in the same place. LOL You are carrying wood in, that has saw dust, bark, and dirt stuck to it. Then you are carrying out ashes that can be still hot. So the best thing it to get the wood stove out of the living area.
As for vacuuming ashes. Most sweepers will not catch all of the dust in there filters. So you still have some fine dust that is airborne. Plus you have the hot ember issue and fire. I have used the vacuum as a dust collector at my ash bucket. I used to put the nozzle under the handle sticking toward the top of the bucket. So when you dumped the ashes into the bucket the vacuum would such in just the dust stirred up. It worked but was more effort than the results made worth while.