The following is Exercise V.3.21 from Hungerford's Algebra (GTM 73).
Let $F$ be algebraic over $K$. $F$ is normal over $K$ if and only if for every $K$-monomorphism of fields $\sigma:F\to N$, where $N$ is any normal extension of $K$ containing $F$, $\sigma(F)=F$ so that $\sigma$ is a $K$-automorphism of $F$. [Hint: Adapt the proof of Theorem 3.14, using Theorem 3.16.]
Theorem 3.14 is the usual characterization of normal extensions. Theorem 3.16 states the properties of normal closures. However, I don't think we need normal closures here.
We shall prove that the above condition implies the following, which will prove that $F$ is normal over $K$: If $\overline{K}$ is any algebraic closure of $K$ containing $F$, then for any $K$-monomorphism of fields $\sigma:F\to\overline{K}$, $\operatorname{im}\sigma=F$ so that $\sigma$ is actually a $K$-automorphism of $F$.
Proof. Extend $\sigma:F\to\overline{K}$ to a $K$-monomorphism $\overline{\sigma}:N\to\overline{K}$ (this is possible since $N$ is algebraic over $F$). Since $N$ is normal over $K$, we have $\operatorname{im}\overline{\sigma}=N$, so $\sigma$ actually maps $F$ into $N$. By hypothesis $\operatorname{im}\sigma=F$, so we're done.
The other direction is rather obvious, and does not need normal closures either. So I have not made any use of normal closures here. But why is the hint asking me to do so? Is my proof correct?
Everything you say seems completely correct, but I think our problem is that the exercise is sloppily stated.
I think that the exercise should be stated as follows: Let $F\supset K$ be an algebraic extension and $N$ a normal extension of $K$ containing $F$. Then $F$ is normal over $K$ if and only if for every $K$-morphism $\sigma:F\to N$, $\sigma(F)=F$.
What he seems to want is that to check for normality of the extension $F\supset K$, you need only embed the situation in some (perhaps well-chosen) normal extension of $K$, not (for instance) an algebraically closed field containing $K$. Then this would be a useful theorem.
Notice that our author has violated the principle of putting all the quantifiers at the beginning. Let this be a lesson to all of us: you’re asking for trouble if you embed quantifiers in the middle of a proposition.