In essence you are describing making a microscope: in the simplest case that is a two-lens system, where the magnification comes about from having a lens near the object (the objective lens) close to the focal distance of that lens, and a second lens (the ocular) collecting that light and focusing it for you (onto your eye, or onto a screen / sensor etc).
The basic equation for a microscope can be found at http://www.schoolphysics.co.uk/age16-19/Optics/Optical%20instruments/text/Microscope_/index.html
In your case, you need to move the ocular lens so that you get a real image instead of a virtual image - in other words, you have to put the objective lens closer to the object than its focal length.
What I just described essentially amount to "taking a photograph of the object with a magnifying glass in front of it", with the added proviso that you can "focus" the image in the camera by playing around with the distance from the magnifying glass to the object. It's much easier to to this by just playing around with the lens rather than computing it - because the actual distance from a lens (especially a compound lens) to a screen / sensor can be quite hard to determine from physical measurements of the lens, but is easy to derive from the optical performance.
Do you know how to take it from there, or do you need more help with the equations?