Easy way and the cleanest way is to purchase a watch pin removal tool
Look at the pins in the link. Line up the pin in the watch to with the pin in the tool and turn the knob until it pushes the pin about a 1/4 of the way out. Then remove with pliers
It's probably best to buy a watch makers hammer too for about $4. Reinsert the pin and use the hammer to tap the pin back in. Take caution and make sure the link is lined up while tapping the pin back in. A whole watch makers tool kit can run $12 and up depending on your needs. I got a great kit off amazon and it has everything