lousy regulation with a resistor in this case
Resistors make terrible current regulators for long chains of LEDs. Escpecially so when you leave only a small left-over voltage to work with. The LED current variation you will expect is:
$$\%I_{_\text{LED}}=\%V_{_\text{LED}}\cdot\frac1{\frac{V_{_\text{CC}}}{V_{_\text{LED}}}-1}$$
So, in your case this would be \$\%I_{_\text{LED}}=-5\cdot \%V_{_\text{LED}}\$.
LED voltages commonly vary by 10%. So this means in your case that could mean 50% variation in the current. (In the opposite direction, so if the LED string voltage increased by 10%, from \$10\:\text{V}\$ to \$11\:\text{V}\$, then the current would decrease by 50%.
This is really very bad for regulation.
On the plus side, using 5 LEDs tends to make the statistical variations a little better as variations of one LED have a chance of being compensated by variations of another one. But it is still quite risky with only 5 LEDs in the string.
use a current source/sink
With that much overhead, though, you can follow the advice given by hacktastical and use that technique for providing a decent current sink/source for the LED chain.
how to design a current sink
By the way, if you are looking to actually learn how to design one of those rather than just being given one -- one that will work in a predictable way over operating temperature and device variations -- then see this link to an EESE answer on this topic.
