First of all, when it comes to cleaning the inside of a wooden watch (the movement), you should always take it to a watch maker. For the outside, it isn’t very complicated. Some people like to carry around a microfiber cloth to clean off smudges. Anything that is good for eye glasses should work perfectly to clean oil and debris off your watch.
If your watch is seriously dirty, then cleaning it with water and liquid hand soap isn’t a problem (assuming it is water resistant to 50 meters or more), liquid soap is better than a bar of soap as any foam and excess soap gets rinsed away. You can gently remove loose debris with a soft toothbrush, make sure it is a soft bristle brush so you don’t scratch the watch. Try to gently submerge the watch into water versus having the faucet water splash on it. That will prevent water from entering worn seals that may have gaskets that have weakened over time. Though, newer dive watches can take a lot of splashing for sure. If you clean the leather strap make sure it fully dries off the wrist before you wear it again otherwise your body heat can make it start to rot!
Some people also like using those ultrasonic jewelry cleaners that use vibrations under water to loosen dirt, but this is only recommended for bracelets once you have removed them from the watch head. You don’t need any fancy chemicals for these machines, just make sure the water is very hot and you have plenty of washing-up liquid to break down any grease. Be aware that oils in the skin soften the edges of any scratches on the glass and these may be more noticeable and look sharper after you clean your watch. In fact, unscrupulous second hand watch sellers used to rub some grease from the sides of their noses over the glass before handing the watch to a client to hide the worst scratches in exactly this way!!!