A fine place for reliable information on the location of the vast majority of cataloged stars is the Simbad Astronomical Database. For example, here is the data for Betelgeuse. As you will see, the first line after "other object types" is the position given in the ICRS coordinate system (the first set of numbers is Right Ascension, in degrees minutes seconds and the second set of number is Declination in degrees minutes seconds). For all but the highest level research-grade astrometry or proper-motion purposes, this position will be more than sufficient.
Now, given you have the 'global' position of the star, you can calculate its relative position as seen from wherever you are on Earth. Doing so is a function of your latitude, longitude and current time. There are online calculators that can do this such as this one, or you can make a program to crunch through the math yourself (it's not exactly trivial, but if there are lots of stars you want to do it for, or you want to do it on the fly, this is a better bet). If you really want to be clever about it and you have lots of stars, you only have to do the calculation once, since all the stars are essentially fixed relative to one another. When all is said and done, the most intuitive local coordinate system to use for your projecting would be altitude & azimuth.
If you are concerned about also including the relative motions of stars over time, I would say don't be. Nearly all stars have proper motions of less than 1 arcsecond a year, which means it would be half a century before most stars have moved in even the most remotely perceptible way to the naked eye (the human eye has an angular resolution of roughly one arcminute). Other apparent motions, such as parallax, are also negligibly small for objects outside of our solar system (again, unless you are doing research-grade work).