The following method should work. The third step is the only one that assumes that $k$ is a finite field; the first two steps only assume that elementary operations in $k$ can be done algorithmically. I have a feeling that one can do better than that, however.
First step: compute a basis of the radical.
This is done by solving a finite system of linear equations given by Dickson's criterion:
An element $r\in A$ is in the Jacobson radical of $A$ if and only if the trace of $\mu_{rs}$ is zero for all $s\in A$, where $\mu_{rs}:A \to A$ is left multiplication by $rs$.
Thus, an element $r$ is in the radical is and only if the trace of $\mu_{ra_i}$ is zero for all $i\in \{1, \ldots, m\}$. If we write $r = \sum_{i=1}^m \lambda_i a_i$, then we want $0=\mu_{ra_j} = \sum_{i=1}^m \lambda_i\mu_{a_ia_j}$ for all $j$. Since the matrix of $\mu_{a_ia_j}$ is easy to compute in terms of the structure constants $c_{i,j,k}$, this gives a system of $m^3$ linear equations with $m$ unknowns $\lambda_1, \ldots, \lambda_m$. You can then algorithmically find a basis $(r_1, \ldots, r_\ell)$ of the set of solutions of this system of linear equations using the method of your choice; this
gives a basis of the radical.
Second step: find the structure constants of the quotient of $A$ by its radical.
Complete $(r_1, \ldots, r_\ell)$ into a basis of $A$ (you can do this algorithmically by looking at $(r_1, \ldots, r_\ell, a_1, \ldots, a_m)$ and finding the leftmost $m$ elements that are linearly independent). Let's say the basis you find is $(r_1, \ldots, r_\ell, r_{\ell+1}, \ldots, r_m)$. Let $\pi:A \to A/J$ be the projection of $A$ to its quotient by its radical. Then $(r_{\ell+1}, \ldots, r_m)$ projects via $\pi$ to a basis of $A/J$.
The matrix of the multiplication by $\pi(r_i)$ in $A$ is obtained by taking the matrix of $\mu_{r_i}$ in $A$, and keeping only rows and columns $\ell+1, \ldots, m$. This defines $A/J$ in terms of the basis elements $\bar r_1 = \pi(r_{\ell+1}), \ldots, \bar r_{m'} = \pi(r_m)$.
Third step: check that $A/J$ is a division algebra.
This is trickier. From now on, let's assume that $k$ is a finite field. Since finite division rings are commutative, the first thing we can do is check that $A/J$ is commutative, simply by checking that all products $\bar r_i \cdot \bar r_j$ commute. If $A/J$ is not commutative, then $A$ is not local.
Assume, then, that $A/J$ is commutative. Let us find its identity element. Consider $\bar r_1$. If $\mu_{\bar r_1}$ is not invertible, then $A/J$ is not a field, so $A$ is not local. If $\mu_{\bar r_1}$ is invertible, then its inverse $\mu_{\bar r_1}^{-1} = \mu_{(\bar r_1)^{-1}}$ is easy to compute in matrix form; applying it to $r_1$, we get an expression of $1$ in terms of the $\bar r_1, \ldots, \bar r_{m'}$.
Thus we can assume, without loss of generality, that $\bar r_1 = 1$. The minimal polynomial of $x_2$ over $k$ is then computable by finding the smallest $d$ such that $1,x_2,x_2^2, \ldots, x_2^d$ are linearly dependent, and by finding a linear dependency for them. Let $P$ be the minimal polynomial. Note that the $k$-span of $1,x_2,x_2^2, \ldots, x_2^{d-1}$ is a sub-$k$-algebra of $A/J$ isomorphic to $k[x]/(P)$.
Over finite fields, algorithms exist to check irreducibility of a polynomial. If $P$ is not irreducible, then $k[x]/(P)$ contains zero-divisors, and so $A/J$ is not a field.
Assume that $P$ is irreducible. Then $L=k[x]/(P)$ is a field contained in $A/J$ as a subalgebra. In particular, $A/J$ is an $L$-algebra. If $A/J = L$, then it is a field, and we conclude that $A$ is local. Otherwise, find an $L$ basis of $A/J$ (it can be constructed from $(\bar r_1, \ldots, \bar r_{m'})$), and repeat the above argument. Since the dimension of $A/J$ over $L$ is strictly smaller than that over $k$, the process will stop at some point.
This may seem like it is little improvement from a brute force check. The reason I think it is better is that the third step works as long as
- $A/J$ is commutative, and
- there exist algorithms to check the irreducibility of polynomials over $k$. This is true for finite fields, for $\mathbb{Q}$, for number fields, ...
