Here's how I would do it:
Install the Dropbox app, if you haven't already done so.
Launch Dropbox, and navigate to the Public folder. Now tap the + at bottom left, and choose Existing Photo or Video. This will launch the photo picker and allow you to select an image which has been saved in your photo library. Once you've done that, the image will be uploaded to your Dropbox account and can be shared or published.
With the photo you just uploaded selected in the Dropbox sidebar (you should see the image on the right), tap the little chain link button in the menu bar, and choose Copy Link to Clipboard. Because you uploaded the image to your Public folder, this will give you a link that you can publish so anyone can see that image file.
Now leave Dropbox and launch Safari. Go to
http://imageshack.us and tap the "url" button next to Upload. Then click in the "Paste image url here" field and paste in the URL that you generated earlier. Now make any changes you want to the image size, label, etc. and upload. When it is done, you'll get a list of links to share your image. I typically use the Forum Code choice. Tap in the field, tap Select All, then tap Copy (which will show up off to the right). Now you can go back to the page where you are editing your post and paste in the contents of the clipboard to insert the image on the spot.