Three comments (which I don't have enough reputation to add as comments):

• The parenthetical claim in the statement of the question is false; Galois groups of local fields need not be supersolvable; for example, $A_4$ is not supersolvable but it is the Galois group of http://www.lmfdb.org/LocalNumberField/2.4.6.7. Another example is $\mathrm{GL}_2(3)$, which is the Galois group of http://www.lmfdb.org/LocalNumberField/2.8.10.2.
• Even if we restrict to extensions of $\mathbb{Q}_p$, giving a precise answer to this question is likely to be difficult. For example, $C_2^4$ is not the Galois group of any extension of $\mathbb{Q}_p$ (it has too many index-2 subgroups), nor is $\mathrm{SL}_2(3)$ (as proved by Weil in his "Dyadic Exercises" paper), even though $\mathrm{GL}_2(3)$ is.
• In general one knows that $G=\mathrm{Gal}(L/K)$ must have a cyclic series of the form $$W\unlhd I\unlhd G$$ in which $W$ is a $p$-group that is also normal in $G$ and $I$ has order prime to $p$ (one can add further constraints imposed by the filtration of $W$ and the action of Frobenius). But as the examples above show, these necessary conditions are not sufficient.

Three comments (which I don't have enough reputation to add as comments):

• The parenthetical claim in the statement of the question is false: Galois groups of local fields need not be supersolvable. For example, $A_4$ is not supersolvable but it is the Galois group of http://www.lmfdb.org/LocalNumberField/2.4.6.7. Another example is $\mathrm{GL}_2(3)$, which is the Galois group of http://www.lmfdb.org/LocalNumberField/2.8.10.2.
• Even if we restrict to extensions of $\mathbb{Q}_p$, giving a precise answer to this question is likely to be difficult. For example, $C_2^4$ is not the Galois group of any extension of $\mathbb{Q}_p$ (it has too many index-2 subgroups), nor is $\mathrm{SL}_2(3)$ (as proved by Weil in his "Dyadic Exercises" paper), even though $\mathrm{GL}_2(3)$ is.
• In general one knows that $G=\mathrm{Gal}(L/K)$ must have a cyclic series of the form $$W\unlhd I\unlhd G$$ in which $W$ is a $p$-group that is also normal in $G$ and $I$ has order prime to $p$ (one can add further constraints imposed by the filtration of $W$ and the action of Frobenius). But as the examples above show, these necessary conditions are not sufficient.

Two comments (which I don't have enough reputation to add as comments):

• The parenthetical claim in the statement of the question is false; Galois groups of local fields need not be supersolvable; for example, $A_4$ is not supersolvable but it is the Galois group of http://www.lmfdb.org/LocalNumberField/2.4.6.7. Another example is $\mathrm{GL}_2(3)$, which is the Galois group of http://www.lmfdb.org/LocalNumberField/2.8.10.2.
• Even if we restrict to extensions of $\mathbb{Q}_p$, giving a precise answer to this question is likely to be difficult. For example, $C_2^4$ is not the Galois group of any extension of $\mathbb{Q}_p$ (it has too many index-2 subgroups), nor is $\mathrm{SL}_2(3)$ (as proved by Weil in his "Dyadic Exercises" paper), even though $\mathrm{GL}_2(3)$ is.

Three comments (which I don't have enough reputation to add as comments):

• The parenthetical claim in the statement of the question is false; Galois groups of local fields need not be supersolvable; for example, $A_4$ is not supersolvable but it is the Galois group of http://www.lmfdb.org/LocalNumberField/2.4.6.7. Another example is $\mathrm{GL}_2(3)$, which is the Galois group of http://www.lmfdb.org/LocalNumberField/2.8.10.2.
• Even if we restrict to extensions of $\mathbb{Q}_p$, giving a precise answer to this question is likely to be difficult. For example, $C_2^4$ is not the Galois group of any extension of $\mathbb{Q}_p$ (it has too many index-2 subgroups), nor is $\mathrm{SL}_2(3)$ (as proved by Weil in his "Dyadic Exercises" paper), even though $\mathrm{GL}_2(3)$ is.
• In general one knows that $G=\mathrm{Gal}(L/K)$ must have a cyclic series of the form $$W\unlhd I\unlhd G$$ in which $W$ is a $p$-group that is also normal in $G$ and $I$ has order prime to $p$ (one can add further constraints imposed by the filtration of $W$ and the action of Frobenius). But as the examples above show, these necessary conditions are not sufficient.