We will prove a more general result by induction (Proof Technique I). Then the theorem will be true as a special case. We will carefully state this result as a proposition indexed by $m$, $m\geq 1$.
$P(m)$: Suppose that $A$ is an $m\times m$ matrix whose entries are complex numbers or linear polynomials in the variable $x$ of the form $c-x$, where $c$ is a complex number. Suppose further that there are exactly $k$ entries that contain $x$ and that no row or column contains more than one such entry. Then, when $k=m$, $\detname{A}$ is a polynomial in $x$ of degree $m$, with leading coefficient $\pm 1$, and when $k<m$, $\detname{A}$ is a polynomial in $x$ of degree $k$ or less.
Base Case: Suppose $A$ is a $1\times 1$ matrix. Then its determinant is equal to the lone entry (Definition DM). When $k=m=1$, the entry is of the form $c-x$, a polynomial in $x$ of degree $m=1$ with leading coefficient $-1$. When $k<m$, then $k=0$ and the entry is simply a complex number, a polynomial of degree $0\leq k$. So $P(1)$ is true.
Induction Step: Assume $P(m)$ is true, and that $A$ is an $(m+1)\times(m+1)$ matrix with $k$ entries of the form $c-x$. There are two cases to consider.
Suppose $k=m+1$. Then every row and every column will contain an entry of the form $c-x$. Suppose that for the first row, this entry is in column $t$. Compute the determinant of $A$ by an expansion about this first row (Definition DM). The term associated with entry $t$ of this row will be of the form \begin{equation*} (c-x)(-1)^{1+t}\detname{\submatrix{A}{1}{t}} \end{equation*}
The submatrix $\submatrix{A}{1}{t}$ is an $m\times m$ matrix with $k=m$ terms of the form $c-x$, no more than one per row or column. By the induction hypothesis, $\detname{\submatrix{A}{1}{t}}$ will be a polynomial in $x$ of degree $m$ with coefficient $\pm 1$. So this entire term is then a polynomial of degree $m+1$ with leading coefficient $\pm 1$.
The remaining terms (which constitute the sum that is the determinant of $A$) are products of complex numbers from the first row with cofactors built from submatrices that lack the first row of $A$ and lack some column of $A$, other than column $t$. As such, these submatrices are $m\times m$ matrices with $k=m-1<m$ entries of the form $c-x$, no more than one per row or column. Applying the induction hypothesis, we see that these terms are polynomials in $x$ of degree $m-1$ or less. Adding the single term from the entry in column $t$ with all these others, we see that $\detname{A}$ is a polynomial in $x$ of degree $m+1$ and leading coefficient $\pm 1$.
The second case occurs when $k<m+1$. Now there is a row of $A$ that does not contain an entry of the form $c-x$. We consider the determinant of $A$ by expanding about this row (Theorem DER), whose entries are all complex numbers. The cofactors employed are built from submatrices that are $m\times m$ matrices with either $k$ or $k-1$ entries of the form $c-x$, no more than one per row or column. In either case, $k\leq m$, and we can apply the induction hypothesis to see that the determinants computed for the cofactors are all polynomials of degree $k$ or less. Summing these contributions to the determinant of $A$ yields a polynomial in $x$ of degree $k$ or less, as desired.
Definition CP tells us that the characteristic polynomial of an $n\times n$ matrix is the determinant of a matrix having exactly $n$ entries of the form $c-x$, no more than one per row or column. As such we can apply $P(n)$ to see that the characteristic polynomial has degree $n$.