I need a new battery for my Apple Watch

For the Apple Watches, there is no way to do this yourself. You can get the battery serviced by Apple. Currently since all Apple stores are closed, you can use your Apple Support App on your iPhone or go to Apples support webpage here. And you will be able to chat with someone and set up a battery replacement. They will send you a box and you place your Apple Watch in it and send it, Apple replaces the battery and sends it back to you. This usually take a few days and possible longer due to current events.

Not sure I you have the actual 1st generation Apple Watch or the S1 model but either case, it's most likely out of warranty so it will cost money to replace it unless it is suffering from a manufacturing defect.

Battery replacement details and costs can be found here. It's a ways down the page under a section called Battery Service. If you are not in the US, I believe you will be able to find the same apple support page for your country by searching for Apple Watch battery replacement on the web.