In short, the narrator is saying
You (the little deer) have just died and gone to paradise. I know you are playing there and ready to run off to your eternal enjoyment -- but hold on a moment -- don't run too far ahead of me: just let me order your grave to honor you, then I'll die as well and join you in the afterlife.
This is possibly confusing for a couple of reasons (outside normal poetic license and archaic language). In particular, "to bespeak thy grave" is to "order a grave for you"; that is "purchase your plot and commission your tombstone or monument".
The following stanza makes this a bit clearer:
First my unhappy statue shall
Be cut in marble, and withal
Let it be weeping too; but there
Th’ engraver sure his art may spare,
For I so truly thee bemoan
That I shall weep though I be stone;
Until my tears, still dropping, wear
My breast, themselves engraving there.
There at my feet shalt thou be laid,
Of purest alabaster made;
For I would have thine image be
White as I can, though not as thee.
That is, "I'm going to order a monument to be placed over your grave; it will be a statue of me, weeping, and standing over your body, carved in alabaster, dead at my feet; but neither my image's tears nor your image's whiteness will do justice to the real thing (my sadness or your purity)."