Using the notion of provability only, how to show that $\Gamma \nvdash \varphi$? For a practical example, suppose I want to show that $\{ P\} \nvdash Q$. From completeness, this is trivial: just find a model where $P$ is true and $Q$ false.
But suppose I am stubborn and I don't want to use completeness. Is there still a way to prove it?
For this particular case, I guess one could find the deductive closure of $\{P\}$, and show that $Q \notin Ded(\{P\})$. But this is simple because we just have one premise. On the other hand, this procedure seems to get complicated in the a general case where we have a set of $n$ premises
$$\varphi_1, \varphi_2 ... \varphi_{n-1}, \varphi_n \nvdash \psi$$
How to proceed then? Then, my question is:
What are the standard ways to show, relying on the notion of provability alone, that
$$\Gamma \nvdash \varphi$$
For some $\Gamma$ and $\varphi$?
 A: There is a way to see that there is no really satisfactory, general answer. Recall that the deduction systems for first-order logic are computably effective. A theory $\Gamma$ is effective if we can computably enumerate its axioms. Because the deductive system is effective, if $\Gamma$ is an effective theory then we can also enumerate its deductive closure $\text{Ded}(\Gamma)$.
But there are many effective theories $\Gamma$ for which $\text{Ded}(\Gamma)$ is not decidable: there is no algorithm to determine, given a sentence $\phi$, whether $\Gamma \vdash \phi$.  One such theory is Peano arithmetic. 
This means that there will be no general, effective way to determine, for an arbitrary theory $\Gamma$ and an arbitrary sentence $\phi$, whether $\Gamma$ proves $\phi$. The spirit of the question is looking for such a method, which presumably would use some kind of syntactic analysis to determine the sentences provable from $\Gamma$. 
For certain particular theories, it is possible to decide whether arbitrary sentences are in the deductive closure. Such theories are called "decidable" in the literature. One example of such a theory is the theory of Boolean algebras. There is a list of theories in the Wikipedia article List of first-order theories, which has several examples of decidable and undecidable theories. Donald Monk's textbook on logic also has detailed coverage of some decidable and undecidable theories, and I recommend it as an entry point to this topic. 
For the theories that are decidable, there are several methods used to establish decidability. Because there is no general method overall, a proof of decidability must necessarily depend on some particular aspects of the theory being studied. One often-used method is known as "quantifier elimination": it is shown that the theory $\Gamma$ proves each sentence $\phi$ is equivalent to some quantifier-free sentence $\phi^{qf}$, and then a syntactic analysis is performed to determine which quantifier-free sentences are provable from $\Gamma$. A particular example of this is Tarski's theorem showing that the theory of real closed fields is decidable. 
