To insert a picture into an online discussion forum posting using the ANGEL HTML editor, a user follows several steps. They click the New Post link and HTML editor link, then click the picture icon to open the upload window. The user browses and selects a photo file, uploads it, and clicks the thumbnail to insert the picture. They can then type their message and click Post to submit. The process involves locating, uploading, and inserting an image file into a discussion post.