Here's the excerpt from the play (Act V, Scene I) that depicts their interaction:
Apothecary: "Who calls so loud?"
Romeo: "Come hither, man. I see that thou art poor.
Hold, there is forty ducats. Let me have
A dram of poison, such soon-speeding gear
As will disperse itself through all the veins
That the life weary taker may fall dead,
And that the trunk may be discharged of breath
As violently as hasty powder fired
Doth hurry from the fatal cannon's womb."
Apothecary: "Such mortal drugs I have, but Mantua's law
Is death to any he that utters them."
Romeo: "Art thou so bare and full of wretchedness,
And fear'st to die? Famine is in thy cheeks,
Need and oppression starveth in thy 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."
Apothecary: "My poverty, but not my will, consents.
Take this potion, if you have the strength of heart
To drink it up."
In this exchange, Romeo does not employ persuasion as much as he appeals to the apothecary's impoverished state and argues that the world offers him no other way to escape his misery. The apothecary, compelled by his dire circumstances, relents and provides Romeo with the poison.
It's worth noting that Romeo's desperation to obtain the poison is driven by his belief that Juliet is dead. He wants to join her in death and sees poison as the only means to achieve that end.