Have Photobucket and this forum open at the same time in different tabs or windows. Log into both.

Now click choose files - this will open up a box so you can browse you computer for images. Select the one you want and click open
Before you do this you may want to change the album (within PhotoBucket) that the image will load into [1] and also set the upload size [2] I would recommend 800x600 as a max for the forum - any bigger and they can be slow to load and too big for some people's screens!

Your photo will upload now, this may take a few moments and will take you to this screen where you can give the photo a title if you want (this can be added later too). Then click save and continue

Now you'll be back in your album. Move your mouse over the image you want to put on the forum.
Next you need to click on the "IMG code" text this will highlight all of the text in the box

Next move your mouse pointer over the highlighted text and Right Click. Choose Copy from the menu
Now go to the forum, start a new post as usual now right-click in the main text box and click paste

This will put the copied text from Photobucket in the post:

Here's what you'll end up with!

Iain