Let $F$ be a left exact functor from the category of sheaves of abelian groups to the category of abelian groups, $\mathscr{F}$ a sheaf of abelian groups on a topological space $X$. Since injective resolutions always exist, and acyclic ones are sent to acyclic ones, we may define the right derived functor $RF$ by $RF(\mathscr{F}):=F(I^\cdot)$ for any injective resolutions $\mathscr{F}\to I^\cdot$.
However, though they exist, an injective resolution is usually messy, so we often do not use it for computations. An example is we use $\check{C}$ech resolution to compute for $F=(f:X\to\{pt\})_*$; note that $R^iF(\mathscr{F})=H^i(X,\mathscr{F})$. More precisely, we pick a good open cover $\mathcal{U}$ for $X$, and then we have $$H^i(X,\mathscr{F}) = \check{H}^{\,i}(\mathcal{U},\mathscr{F}) \mbox{ [Harshorne A.G. ex.III.4.11].}$$ For good enough spaces, we can pick such a good cover and compute the right-hand side precisely.
Questions:
In general is there a way to compute a derived functor first by resolving by a Cech complex with a good cover?
If we cannot expect my first question to be true, is it at least possible for some specific functors?
When $F=(f:X\to Y)_*$, we have a clearer description: $R^iF(\mathscr{F})$ is the sheaf that associates to the presheaf $$V\mapsto H^i(f^{^1}(V), \mathscr{F}|_{f^{-1}(V)}) \mbox{ [Harshorne A.G. III.8.1].}.$$ Good! this makes things more explicit! I notice that this result can be obtained by my first question (if it is true). However, Hartshorne uses a quite complicated proof that refers to other concepts such as "effacable", "universal $\delta$-functors". I also found that if I prove it directly by definition, it will be a mess (Homology sheaf is a quotient by the image sheaf.. so you have to take two sheafifications!). Is there a plain explanation?
For the third question, is it possible to get an explicit result just for $RF(\mathscr{F})$ but not $R^iF(\mathscr{F})$? I would like to know since $RF(\mathscr{F})$ contains more information.
1) : This is true and is a consequence of the Čech-to-derived functor spectral sequence (see the wikipedia page, where they mention that we can use this to compute derived functors).
2) : see 1)
3) I'm not sure I really understand this part, but taking a (or two) sheafification is not really a big deal, and indeed this description is really useful in particular when you are taking the stalks.
4) You can't obtain $RF(\mathcal F)$ unless some really special case, since you would need an explicit resolution which is hard to produce. However there is a case when you can compute it : Deligne's highly non-trivial theorem tells you that $$Rf_* \underline{\Bbb Q}_X = \bigoplus_i R^if_*\underline{\Bbb Q}_X[-i]$$ when $f : X \to Y$ is a proper morphism between projective smooth varieties. Concretely this means that $Rf_* \underline{\Bbb Q}_X$ is isomorphic as a complex to its cohomology in the right degree, with zero differential. The proof uses tools from Hodge theory.
Additional remarks :
a) Let me add that projective (i.e by locally free sheaves) resolutions are more easy to find, and you can compute the full complex $R\text{Hom}$ or the derived tensor product very explicitely. For example if $X \subset \Bbb P^n$ is an hypersurface, the exact sequence $$ 0 \to \mathcal{O}_{\Bbb P^n}(-X) \to \mathcal{O}_{\Bbb P^n} \to \mathcal{O}_X \to 0$$ can be interpreted as a $2$-step projective resolution of the coherent sheaf $\mathcal O_X$. Since $\text{Hom}$ is a bifunctor, you can choose to take a projective resolution, rather than an injective one :-)
b) In the derived setting, you rarely get something explicit for free, and the main (only?) tools available are spectral sequences.
c) An excellent reference is The Fourier-Mukai transfom in algebraic geometry which contains a very clear exposition of derived functors and the derived category of coherent sheaves, assuming a minimal background in homological algebra (you probably need to know what is a spectral sequence though, but if you want to understand the details you can't skip it).