You can't delete individual messages, but you can delete entire conversations, if you want to clean up your inbox.
From the Chat page, move your cursor over the list of partners, to the left. A small three-dotted button will show up on that row. Clicking on it will allow you to either Delete this conversation or Block that person entirely.
You can do this on mobile as well, but with an extra step. Once in your Chat page, on the row where the partner you want to remove is, press and hold. After a second, the button will show up on your screen and you can tap it to open the same menu.