As Weinberg exposited in his QFT Vol1, there are two equivalent ways of arriving at the same quantum field theories:
(1). Start with single-particle representations of Poincare group, and then make a multiparticle theory out of it, while preserving principles of causality etc. I would call this approach the second quantization of particles, since second quantization is usually used to emphasize the many-body nature of a theory.
(2). Start with field representations of Poincare group, canonically quantize it, while preserving principles of causality, positive definiteness of energies etc. I would call this approach the quantization of fields, just as everyone else would call it.
Weinberg showed the proof of the equivalence between the above two approaches using some, though not hard, but let's say nontrivial, mathematics. The equivalence seems like a sheer miracle to me, or a complete coincidence. I do not feel that I understand the equivalence with the current state of mind. Is there a way to trivialize the equivalence? Or putting it another way, is there an a priori reasoning to argue, given the two sets of starting points of (1)(2), we have to get the same theory in the end?
Just as a side remark, many have suggested the term "second quantization" should be totally dumped, because it is really just the first quantization of fields. To me however, it still serves some purposes since the equivalence is not transparent.
Answer
The presumed equivalence between the canonical quantization and the Fock space representation is only a particular case.
The canonical formalism provides only with canonical Poisson brackets. The first step according to Dirac's axioms is to replace the Poisson brackets by commutators and since these commutators satisfy the Jacobi identity, they can be represented by linear operators on a Hilbert space.
Canonical quantization does not specify the Hilbert space.
Finding a Hilbert space where the operators acts linearly and satisfy the commutation relations is a problem in representation theory. This task is referred to as "quantization" in the modern literature.
The problem is that in the case of free fields, this problem does not have a unique solution (up to a unitary transformation in the Hilbert space). This situation is referred to as the existence of inequivalent quantizations or inequivalent representations. The Fock representation is only a special case. Some of the quantizations are called "non-Fock", because the Hilbert space does not have an underlying Fock space structure (i.e., cannot be interpreted as free particles), but there can even be inquivalent Fock representations.
Before, proceeding, let me tell you that inequivalent quantizations may be the areas where "new physics" can emerge because they can correspond to different quantum systems.
Also, let me emphasize, that the situation is completely different in the finite dimensional case. This is because that due to the Stone-von Neumann theorem, any representation of the canonical commutation relations in quantum mechanics is unitarily equivalent to the harmonic oscillator representation. Thus the issue of inquivalent representations of the canonical commutation relations occurs only due to the infinite dimensionality.
For a few examples of inquivalent quantizations of the canonical commutation relations of a scalar field on a Minkowski space-time, please see the following article by: Moschella and Schaeffer. In this article, they construct inequivalent representations by means of Bogoliubov transformation which changes the vacuum and they also present a thermofield representation. In all these representations the canonical operators are represented on a Hilbert space and the canonical commutation relations are satisfied. The Bogoliubov shifted vacuum cases correspond to broken Poincare' symmetries. One can argue that these solutions are unphysical, but the symmetry argument will not be enough in the case of quantization on a general curved nonhomogenous manifold. In this case we will not have a "physical" argument to dismiss some of the inequivalent representations.
The phenomena of inequivalent quantizations can be present even in the case of finite number of degrees of freedom on non-flat phase spaces.
Having said all that, I want nevertheless to provide you a more direct answer to your question (although it will not be unique due to the reasons listed above). As I understand the question, it can be stated that there is an algorithm for passing from the single particle Hilbert space to the Fock space. This algorithm can be summarized by the Fock factorization:
$$ \mathcal{F} = e^{\otimes \mathcal{h}}$$
Where $\mathcal{h}$ is the single particle Hilbert space and $\mathcal{F}$ is the Fock space. As stated before canonical quantization provides us only with the canonical commutaion relations:
$$[a_{\mathbf{k}}, a^{\dagger}_{\mathbf{l}}] = \delta^3(\mathbf{k} - \mathbf{l}) \mathbf{1}$$
At this stage we have only an ($C^{*}$)algebra of operators. The reverse question about the existence of an algorithm starting from the canonical commutation relations and ending with the Fock space (or equivalently, the answer to the question where is the Hilbert space?) is provided by the Gelfand -Naimark-Segal construction (GNS), which provides representations of $C^{*}$ algebras in terms of bounded operators on a Hilbert space.
The GNS construction starts from a state $\omega$ which is a positive linear functional on the algebra $ \mathcal{A}$ (in our case the algebra is the completion of all possible products of any number creation and annihilation operators).
The second step is choosing the whole algebra as an initial linear space $ \mathcal{A}$. In general, there will be null elements satisfying:
$$\omega (A^{\dagger}{A}) = 0$$
The Hilbert space is obtained by identifying elements differing by a null vector:
$$ \mathcal{H} = \mathcal{A} / \mathcal{N} $$
($\mathcal{N} $ is the space of null vectors).
The inner product on this Hilbert space is given by:
$$(A, B) = \omega (A^{\dagger}{B}) $$
It can be proved that the GNS construction is a cyclic representation where the Hilbert space is given by the action of operators on a cyclic "vacuum vector". The GNS construction gives all inequivalent representations of a given $C^{*}$ algebra (by bounded operators). In the case of a free scalar field the choice of a Gaussian state defined by its characteristic function:
$$ \omega_{\mathcal{F} }(e^{\int\frac{d^3k}{E_k} z_{\mathbf{k}}a^{\dagger}_{\mathbf{k}} + \bar{z}_{\mathbf{k}}a^{\mathbf{k}} }) = e^{\int\frac{d^3k}{E_k} \bar{z}_{\mathbf{k}} z_{\mathbf{k}}}$$
Where $z_{\mathbf{k}}$ are indeterminates which can be differentiated by to obtain the result for any product of operators.
The null vectors of this construction will be just combinations vanishing due to the canonical commutation relations (like $a_1 a_2 - a_2 a_1$). Thus this choice has Bose statistics. Also subspaces spanned by a product of a given number of creation operators will be the number subspaces.
The state of this specific construction is denoted by: $\omega_{\mathcal{F}}$, since it produces the usual Fock space. Different state choices may result inequivalent quantizations.
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