There are two ways that I know of for uploading a PDF directly from your computer.
Option 1 - simple and fast
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On the top of the reply box is an icon of a picture. Clicking it will allow you to browse and attach files from your PC. Even know it says it's for pictures, it will also upload PDFs.
Make sure you have the "From Computer" tab selected, not the "From URL" tab. The URL tab is the default tab whenever you open that window, so you will have to select the one for uploading from your computer. Once you select the file you want, click "Upload Files" and you're done.
Skeeter Skelton - My Friend the 357.pdf
Option 2 - A little more complicated, but useful for posting and managing numerous attachments at once.
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If you click "Go Advanced" on the quick reply box, it will open up an expanded reply box with a number of additional options. One of them being for attachments (the paperclip)
Clicking it will open a new window that has options for any attachments you would like to upload. Clicking "Add Files" will will allow you to browse and add files, including PDFs. Once your file(s) are chosen, click "upload".
Once it uploads (depending on the file size and your internet speed, it might take awhile), it should automatically be "checked". If not, check it and click "Insert Inline" if you want to have the file somewhere in the middle of your post, and click "done". Alternatively, you can just click "Done" without inserting the file into your post. This will cause it to appear at the end of your post as an attachment.
Skeeter Skelton - My Friend the 357.pdf
I hope this helps.