I need a good reference book where I can learn the cohomology of sheaves through the approach of Čech cohomology. The Hartshorne's book, for example, doesn't help me a lot because he choose the "derived functors approach".
The number one account is still in Serre's legendary Faisceaux Algébriques Cohérents (Chapitre I,§§3,4) of which you can find an English translation here.
You can also look at sections 7.8 and 7.9 of Taylor's textbook .
Another excellent textbook is that by Fritsche-Grauert, where you will find in Chapter IV,§3 not only Čech cohomology for sheaves but also its relation to classical singular cohomology.
A more technical account for algebraic geometers is in Chapter VII of Mumford-Oda's notes .
The most "down-to-earth" book I know that covers this topic is Rick Miranda's Algebraic Curves and Riemann Surfaces. It goes slow, has a lot of examples, and has minimal prerequisites.
You might also try Bott and Tu's Differential forms in Algebraic Topology or Bredon's Topology and Geometry for different perspectives.