I am trying to calculate the HOMFLY polynomial for the oriented Hopf link.
I have a left handed corkscrew with linking number $= -1$ (see drawing below).
Using the equation $$lP_{H^{+}} + l^{-1}P_{H^{-}} + mP_{H_{0}} = 0$$
I understand what the diagrams of the positive and negative crossings will look like, and that $P_{H_{0}}$ will equal $P_o(\text{the unknot})$ and that $P(\text{unknot})= 1$ but I'm not sure how to turn my equation using diagrams into an equation of two variables.
First of all, it looks like there's a typo in the question, and that the second $l$ should have been an $l^{-1}$.
Now, we claim that the HOMFLY polynomial of a two-component unlink is $-(l+l^{-1})/m$. To see this, add a crossing connecting the two components. Then, no matter if the crossing is positive or negative, the result is an unknot, so it follows from the skein relation that
$$lP(\text{Unknot}) + l^{-1}P(\text{Unknot}) + m P(\text{Two-component unlink}) = 0,$$
from which the claim follows. That is, in pictures,
Moving on to the Hopf link, let's focus on the topmost (negative) crossing.
Resolving that one into a positive crossing will leave you with a two-component unlink.
Resolving it into a non-crossing leaves you with an unknot.
Now, by the skein relation
$$l P(\text{Two-component unlink}) + l^{-1}P(\text{Hopf link}) + mP(\text{Unknot}) = 0$$
or, once again in pictures,
Rearranging things a bit, we find that
$$P(\text{Hopf link}) = -l(l P(\text{Two-component unlink}) + mP(\text{Unknot})) = -l(l(-(l+l^{-1})/m) + m) = l^3m^{-1} + lm^{-1} -lm.$$