It's worth noting that Juliet's "death" isn't really her death at all in this context. Instead, she has taken a strong sleeping potion that makes her appear dead in an effort to escape and live with Romeo. Friar Lawrence actually pens a letter to Romeo to explain this scheme to reunite the young lovers, but in perhaps the most tragic turn of events in the play, the letter never makes it to him.
News quickly spreads of beautiful Juliet's "death," and Balthasar, Romeo's servant, goes to Mantua to tell Romeo of this tragic turn of events. Romeo is, of course, immediately and visibly upset. Balthasar notes:
I do beseech you, sir, have patience.
Your looks are pale and wild, and do import
Some misadventure. (V.i.27-29)
Romeo asks if Balthsar has a letter for him, but he finds a dead end there. He therefore makes the immediate decision to find an apothecary and purchase a poison to end his own life. This sets in motion a series of events that will lead Juliet to her real and actual death.