thonet restoration  

The skills and knowledge necessary for
responsible furniture restoration are not something one can pick up by spending a weekend on the web. Professionals spend lifetimes learning it.
Your chair is particularly problematic in that it has seen hard use, neglect, and features dissimilar materials, each requiring different approaches to treatment. For instance, the basic construction is cellulose, the padding and covering could be cellulose, protein, polymers, or a combination, and the hardware is metallic. Without disassembly, restoration of these individual components is practically impossible. Many would consider the effort not worth the time and expense.
If the chair were mine, I'd give it a good going over with a weak solution of distilled water and a mild laundry detergent like Ivory liquid. Avoid hardware store variety "miracle" products or solvents. Use clean rags and a soft toothbrush to work at the buildup of grime between the tacks. Clean a manageable area at a time and don't allow the surfaces to soak up a lot of water. Generously overlap each area and try not to "overclean" and end up with a splotchy appearance. Rinse and wipe dry as you go. This should take a good hour or two of fairly intense effort.
Assess your results after a complete surface cleaning. These processes can be as involved and labor-intensive as you want to make them. Some find them satisfying. Others might see them more pragmatically. It depends on one's personality and goal, I think.
Check in here once that's done with your thoughts on the experience and/or for advice on the next steps to consider.
Bon Chance!