If $A$ and $B$ have characteristic polynomial $x^5-x^3$ and minimal polynomial $x^4-x^2$, then they're similar. Let $A$ and $B$ be matrices with coefficients in some field $k$.  If they both have characteristic polynomial $x^5-x^3$ and minimal polynomial $x^4-x^2$, then they must be similar.
I've thought about this for awhile now, and I still have not made any progress.
 A: Do you know about Jordan Canonical Form?
Two matrices are similar if and only they have the same Jordan Canonical Forms (same meaning up to rearrangements of the Jordan Blocks).
So we first factor $x^5 - x^3 = x^3 \cdot (x-1)(x+1)$ which means they have eigenvalues (including multiplicity, obviously) $0, 0, 0, 1, -1$. 
If your field is not of characteristic $2$:
Since $1$ and $-1$ appear in the characteristic polynomial once, their Jordan blocks must have size $1$.$0$, a priori, could have a block of size $3$, a block of size $2$ and a block of size $1$, or three blocks of size $1$.
If your field is of characteristic $2$:
Your eigenvalues are $0, 0, 0, 1, 1$. Block sizes for $0$ are the same, block sizes for $1$ are one block of size $2$ or two blocks of size $1$.
We factor their minimal polynomial as $x^4 - x^2 = x^2(x+ 1)(x-1)$. The minimal polynomial tells us the largest Jordan Block corresponding to the eigenvalue $0$ is of size exactly $2$.  If we have one Jordan block of size exactly $2$, the other Jordan block is determined to be $1$. 
If your field is of characteristic $2$, then you also get that the block with ones on the diagonal must be of size $2$ since the factor appears twice in the minimal polynomial.
So there is only one option for the Jordan Canonical form of a matrix with this characteristic polynomial and minimal polynomial, 
In characteristic not $2$, a block of size $1$ with $1$ on the diagonal, a block of size $1$ with $-1$ on the diagonal, a block of size $2$ with $0$ on the diagonal, and a block of size $1$ with $0$ on the diagonal. 
In characteristic $2$, a block of size $2$ with $1$ on the diagonal, a block of size $2$ with $0$ on the diagonal, and a block of size $1$ with $0$ on the diagonal.
As there is only one option for the Jordan Forms, they must have the same Jordan form, hence are similar.
