Maybe you already know of the notes by Benoît Audoubert and Orsola Tommasi entitled "Mixed Hodge Structures," notes on Audoubert's seminars on Mixed Hodge structures at the University of Nijmegen in 2002? These notes certainly discuss quasi-projective algebraic varieties (in Section 3). It is about 50 pages long, with the following (high-level) Table of Contents:

1. Hodge structures
2. Varieties with normal crossings
3. Smooth quasi-projective varieties
4. The theory of Deligne