Rumiko, would you go on a date with me this weekend?
Literal
Rumiko, this weekend [at-に] me [with-と] date do [won't-you-てくれませんか]?
~てくれませんか (won't you do X for me?) is a polite request that foregrounds the speaker's desire for the listener's cooperation. ボク is a casual male first-person pronoun written in katakana for stylistic effect — common in manga and casual writing. デートする is a loanword verb.