How does Romeo convince the apothecary to sell him poison?

The answer to this can be found in Act V Scene 1. 
Basically what Romeo does is he tells the apothecary "you are so poor, you really need
to sell this stuff to me so you will not starve."

That is
why Romeo picks this particular apothecary to buy the poison from.  He looks around to
find an apothecary who looks like he is really poor and does not have much luck selling
stuff to people.

He tells the apothecary that he can see
that the guy is terribly poor and hungry.  That's why the man is willing to
sell.