Step 1: Create an Automator folder action to copy all files added to your hidden Photo Stream folder to a folder elsewhere on your computer.
Start in Automator by creating a 'Folder Action' workflow. Click 'Choose folder', then press SHIFT-COMMAND-PERIOD while the Automator folder picker is open so that you can browse hidden folders. Choose the iCloud Photo Stream folder here: {USERNAME}/Library/Application Support/iLifeAssetManagement/assets/sub
Add a 'Get Folder Contents' action to your workflow, being sure to check the 'Repeat for each subfolder found' option.
Add a 'Copy Finder Items' action to your workflow, selecting a folder where you'd like Lightroom to import from. Something like "/Pictures/Lightroom Import" might be a good idea.
Save your workflow. You've just activated an Automator action that copies new Photo Stream images to a folder of your choosing.
Step 2: Tell Lightroom to automatically import images from this folder. In the Lightroom menu go to: File -> Auto Import -> Auto Import Settings... This will move the files out of the folder you just created and into Lightroom, leaving the hidden photostream directory un-modified so that we don't confuse iCloud or prevent you from viewing your Photo Stream on other iOS devices.
Step 3: There is no step 3. Just go out and take some awesome photos. Next time you launch Lightroom, they'll be there, and your images will continue to be imported on the fly when Lightroom is open.