$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sym{Sym}$Question: "Which representations of $\mathfrak{sl}(2)$ are homomorphic images of the tensor product of finitely many copies of $\mathbb{C}^2$?"
Answer: If $k$ is the field of complex numbers and if $W:=k\{e_1,e_2\}$ it follows that any finite dimensional irreducible $\SL(W)$-module $V$ decomposes as
$$V \cong \bigoplus_i \Sym_k^{m_i}(W),$$
where $m_i\geq 1$ are integers. There is for every $i$ a surjection
$$\pi_i: W^{\otimes m_i} \rightarrow \Sym_k^{m_i}(W)$$
hence you get a surjection
$$\pi: U:=\bigoplus_i W^{\otimes m_i} \rightarrow V \cong \bigoplus_i \Sym_k^{m_i}(W).$$
Hence every module $V$ is a quotient of a direct sum of tensor powers of $W$.
I'm unsure if you can choose $U$ to be a tensor product of $W$ — you may need to take direct sums.
In Fulton–Harris's book "Representation theory - a first course", §15.3, they give an elementary construction of all finite dimensional irreducible $\SL(n,k)$-modules as submodules of the tensor product
$$\Sym^{a_1}(V)\otimes \Sym^{a_2}({\bigwedge}^2 V) \otimes \dotsb \otimes \Sym^{a_{n-1}}({\bigwedge}^{n-1} V)$$
where $V:=k^n$, by giving an explicit construction of the highest weight vector. In the general case you must use tensor and exterior products to get all representations.
Note 1: For $\SL(2,k)$ it follows $W \cong W^*$ is an isomorphism, hence
you get isomorphisms of $\SL(2,k)$-modules
$$W^{\otimes m} \cong \operatorname{End}_k(W)^{\otimes n}$$
if $m=2n$ and
$$W^{\otimes m} \cong \operatorname{End}_k(W)^{\otimes n}\otimes W$$
if $m=2n+1$.
Note 2: On page 472 in FH you find the following construction in characteristic zero: There is for any $n,d\geq 1$ a split surjection
$$ V^{\otimes dn} \xrightarrow p \Sym^n({\bigwedge}^d V)$$
and if $G:=\SL(V)$ it follows $p$ is a split surjective map of $G$-modules.
Using this construction you get for any integers $a_1,\dotsc,a_{n-1}$ a split surjection of $G$-modules
$$ V^{\otimes \sum_i ia_i} \rightarrow \bigotimes_i \Sym^{a_i}({\bigwedge}^i V)$$
and since any finite dimensional irreducible $G$-module $W$ is a submodule of
$$\bigotimes_i \Sym^{a_i}({\bigwedge}^i V)$$
it follows you get a split surjection of $G$-modules
$$V^{\otimes \sum_i ia_i} \rightarrow W$$
for any finite dimensional irreducible $G$-module $W$. Hence in characteristic zero any finite dimensional irreducible $W$ may be realized as a submodule (or quotient module) of $V^{\otimes n}$ for some integer $n$. Hence if you want to give an explicit and elementary construction of all finite dimensional irreducible $\SL(V)$-modules, you may use the tensor powers $V^{\otimes n}$ for all $n\geq 1$. The above argument gives a construction of a highest weight vector
$$ v_{\lambda} \in V^{\otimes n}$$
for any finite dimensional irreducible $SL(V)$-module $V(\lambda)$.
Example: Given $W_1\subseteq V^{\otimes n_1}, W_2 \subseteq V^{\otimes n_2}$. Let $V:=\mathbb{C}^{n}$ and let $T:=\wedge^n V$ be the trivial module. You may construct $T \subseteq V^{\otimes n}$ as a sub-module. For $W_1\oplus W_2$ to be realized as a quotient/submodule of $V^{\otimes d}$ for some $d$ there are obvious "obstructions". If $n_1,n_2$ are even and $n$ is odd there is no common tensorpower $V^{\otimes d}$ containing the module
$$W_1 \otimes T^{\otimes i}\text{ and }W_2\otimes T^{\otimes j}$$
for integers $i,j$. This is similar to the case of $SL(\mathbb{C}^2)$.
"Thank you so much for the reference to that fact in Fulton/Harris. I will check it out. – Malkoun 18 hours"
The above construction gives an "elementary" construction of all finite dimensional irreducible $\SL(V)$-modules by constructing an explicit highest weight vector $v\in V^{\otimes n}$ for some $n \geq 1$. I believe one of the reasons for introducing the Schur-Weyl functors in FH is to relate this study to the study of the representations of the symmetric group and combinatorics.
"I am still reading parts of Fulton and Harris. I get that such "tensorial" constructions can produce irreducible representations for the An type Lie groups, via the works of Schur, Weyl, Young etc. I wonder if similar constructions exist for all classical groups (I suspect the answer is yes). But what about the 5 exceptional cases? Is there a known general explicit construction for the highest weight representations which includes such "tensorial" constructions, but also works for the exceptional Lie groups? If this is known, then this would probably be directly relevant to my 2 projects! – Malkoun"
Dixmiers book "Enveloping algebras" give a construction of all finite dimensional irreducible $\mathfrak{g}$-modules for any semi simple Lie algebra using the universal enveloping algebra $U(\mathfrak{g})$ and Verma modules.
Any finite dimensional irreducible $\mathfrak{g}$-module $V(\lambda)$ has a "canonical" highest weight vector $v$ and you get an exact sequence
$$0 \rightarrow I(\lambda) \rightarrow U(\mathfrak{g}) \rightarrow^{\rho_v} V(\lambda) \rightarrow 0$$
where $v\in V(\lambda)$ is a highest weight vector for $V(\lambda)$ and the map $\rho_v$ is defined by $\rho_v(x):=xv$. The ideal $I(\lambda)$ is the (2-sided) "annihilator ideal" of the vector $v$. In Theorem 7.2.6 in Dixmier they describe all finite dimensional irreducible $\mathfrak{g}$-modules by giving an explicit 2-sided maximal ideal $I(\lambda) \subseteq U(\mathfrak{g})$. In Proposition 7.2.7 they give generators of $I(\lambda)$.