During one of my lectures my teacher defined the characteristic of a field $F$ as the generator of the kernel of the only homomorphism $\iota: \mathbb{Z}\to F$ that maps $0_{\mathbb{Z}}$ to $0_{F}$ and $1_{\mathbb{Z}}$ to $1_F$. If $F$ is a field, either $\text{char}(F)=0$ or $\text{char}(F)$ is prime because $\mathbb{Z}/\text{ker}(\iota)\cong F$. I suppose he claims this referring to the fundamental homomorphism theorem together with the assumption that $\iota$ is surjective. Why should it be so? The claim seems generally false.
Does the isomorphism claim hold? If it does, for which reasons? If not, can we modify the assertion somehow to make it true?
The claim is false. Indeed the homomorphism clearly isn't surjective in the characteristic 0 case, and it isn't always surjective in the characteristic $p>0$ case either. $\mathbb{Z}/\ker(\iota)$ is isomorphic to the characteristic subfield, in fact this is a possible definition of the characteristic subfield.
The reason we know that $\ker(\iota)=p\mathbb{Z}$ for a prime $p$ is that the image of the homomorphism is a subring of a field, and hence a domain. Thus the kernel is prime in $\mathbb{Z}$, and it can be checked that in a PID like $\mathbb{Z}$, the prime ideals are maximal, and thus $\mathbb{Z}/\ker(\iota)$ is a field.
An example where the homomorphism is not surjective in the characteristic $p>0$ case would be with fields such as $\mathbb{F}_{p^2}$, like $\mathbb{F}_2[x]/(x^2+x+1)$.
Also I feel I should add that $GL_2(F)$ is not a field as the comments have mentioned.