The speaker of this poem is a little dude named Astrophel. He's a very gifted poet (as you no doubt can tell), and he really wants this girl named Stella. The problem is she doesn't want him; in fact, she's planning on marrying another guy. Astrophel, whose name means "the one who loves a star," is a persistent guy and refuses to give up. He's also a dreamer, both literally and figuratively. He has grand plans for him and Stella, but he also has very vivid dreams about his love. This is why he's so desperate to get to sleep: because he wants to see the glittering, lively image of Stella that he describes in the sonnet that precedes "Come Sleep! O Sleep."
We know the speaker of this poem is Astrophel, but we can't help thinking he reminds us just a little bit of the guy who wrote this poem: Sir Philip Sidney. Just to clarify, the speaker is not Sidney (that's always a dangerous assumption in poetry), but he kind of resembles him a little bit. Like Astrophel, Sidney too was (supposedly) in love with a woman who didn't love him back (Lady Penelope Devereux). Sidney too was a dreamer, the "flower of chivalry" (as he was dubbed), a legendary courtier (i.e., a dude who hangs around the court of the King and/or Queen), a romantic… basically everything a sonneteer should be. Right?
Even if you knew nothing about Sidney, and nothing about Astrophel, you could still determine a few things about the speaker of this poem. He's really tired—that's for sure. He wants to sleep, largely because he's been dealing with a lot of issues: despair, emotional conflicts, things that make you want to sleep so you can stop dealing with it. And here's the other thing about the speaker: he's willing to make some sacrifices just to get some sleep. Recall that he offers Sleep a bedchamber, pillows, and a nice bed in exchange, and even a glimpse of the mental image he has of the oh-so-beautiful Stella. That's the other thing, actually; the speaker is a lover and a romantic if we ever saw one. Sleep isn't just about rest for this guy; it's about getting to spend time with Stella in a dreamy fantasy that may never come true. Oh Sidney, you're making us blush.