You can set up an e-mail signature in Office365's OWA (also works with on-premise Exchange 2013 OWA) quite easily. While it's no problem to apply any kind of formatting to your signature, you'll find out that adding images is not as straight forward.
To set up an e-mail signature, follow the steps below.
For best results, use either Internet Explorer or Firefox. Google Chrome has some (minor) issues with OWA.
When attempting to copy/paste an image in the signature, at first, it will seem to work when saving the image. But when creating a new mail, you'll notice the image is gone. This is because signatures in OWA can only work with externally linked images; inline images are not supported. Copy/pasting an image directly will link that image to a temporary URL. When creating a new e-mail, that temporary location will no longer be valid, and the image will therefore appear empty.
The solution is to place the image on a webserver somewhere (one that is permanently available), then refer to that image. As long as the webserver and image are available, the image will be available in your signature as well.
It's worth noting that because images through OWA are linked images and never inline, the images may be filtered out by the recipient's e-mail program - Outlook is known to do this for any sender not in your address book.