AO3 doesn't have native image hosting so you'll need to host your image on another site before you can post your art. In this guide, I'll show you how to use Imgur, Tumblr, and LiveJournal as image hosts. You can click to enlarge the images.
Hosting on Imgur
Step 1: Go to Imgur and click "New Post".
Step 2: Browse your computer or drag and drop your image onto Imgur.
Step 3: You'll be taken to a page similar to the one below. Imgur provides image links but AO3 doesn't display those images in my experience. You'll need a direct link in order to post. Right-click on your image and select "Copy image address". This will give you the direct link and you can move on to part 2 from here.
Getting Links with an Imgur Account
Step 1: If you have an Imgur accout and you've already posted your images, you can hover over your name in the top right corner and select "images". You'll see your images displayed like the image below.
Step 2: Select your desired image and you'll get a page with a list of links. Copy the direct link.
Hosting on Tumblr
Step 1: Click the photo icon at the top of your dash and then select or drag your image into the window that pops up. Make sure that you select "Post privately" before you post the image.
Step 2: Navigate to your posts. You can do this by clicking the person icon in the right hand corner and selecting "Posts" or you can replace your username in this URL: https://www.tumblr.com/blog/yourusername
Step 3: Your private post should be there. Right-click the image and select "Copy image address".
Hosting on LiveJournal
Step 1: Hover over your username and select "Scrapbook".
Step 2: Click on "Upload Photo". Then select or drag images into the pop up window. You'll need to name an album if you don't already have one.
Step 3: Select your image and then right-click to select "Copy image address".