How does Romeo convince the apothecary to sell him poison?
Romeo's task in convincing the apothecary to sell him the poison is an arduous one as both he and the apothecary know that the sale of such posion is illegal.
In fact, when Romeo inquires about such poison, the Apothecary replies:
Romeo takes one look at the Apothecary and the nature of his existence and then makes his appeal:
famine is in thy cheeks,
Need and oppression starveth in thine eyes,
Contempt and beggary hangs upon thy back;
The world is not thy friend nor the world's law;
The world affords no law to make thee rich;
Then be not poor, but break it, and take this.
Because the Apothecary is poor and starving, Romeo appeals to his most basic need--money. And the Apothecary responds:
In other words, the Apothecary knows it is wrong to sell the poison and infers that Romeo intends to do harm with it, but ultimately he is so desperate for money that he is willing to take those risks.
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.