Write the given concentration.

The concentration is $0.0006214 \text{ g/ml}$.

Move the decimal point 4 places to the right to get a number between 1 and 10.

Let $N = 6.214 \times 10^{-4}$. We want to find $n$ such that $N = 10^n$.
Taking the base-10 logarithm of both sides:

Using the logarithm property $\log(AB) = \log A + \log B$:

From logarithm tables:
The logarithm of 6.21 is 0.7931.
The mean difference for 4 in the row for 6.21 is 3.
So, $\log_{10}(6.214) = 0.7931 + 0.0003 = 0.7934$.

Substitute the value of $\log_{10}(6.214)$ into the equation for $n$:

The concentration is $10^{-3.2066} \text{ g/ml}$.

The concentration expressed in the form $10^n$ is $\boxed{10^{-3.2066} \text{ g/ml}}$.