How would I rotate the view port along the axis from my forehead to the monitor, like as if I was tilting my head (but more like tilting the viewport)
It looks like you'd like to avoid viewport rotation lock around Z axis and to have possibility to rotate freely with RMB.
Option 1 - switch to Trackball rotating method
By default, Blender uses Turntable rotation method for manipulating view. It means the rotation in 3D Viewport will be locked to Z axis. See what the difference between Trackball and Turntable orbit styles.
To change that in File > User Preferences (or press Ctrl+Alt+U) choose Input page and find Orbit Style part.
After toggling it to Trackball the 3D Viewport will rotate freely.
Option 2 - use View Roll
Without changing anything in preferences it's possible to roll the view to change its angle. Hold Ctrl+Shift and rotate the mouse wheel
(or by using the key shortcuts Shift+Pad4 / Shift+Pad6) to rotate left / right: