No, she is not.
Hatsune Miku's is a character that is a personification of her Vocaloid 2 voice bank, she was designed and drawn by KEI (a Japanese illustrator). The character is owned by Crypton Future Media. Her voice was provided by Saki Fujita, a voice actress.
She has grown to be a noticeable Vocaloid and has been cosplayed numerously, she also had a concert in which a projected rendering displays her image as if she were on stage.
If you mean "is Hatsune Miku real", then one would have to define the meaning of "real". She is real in the sense that yes, you can go out and buy her, and use her in your own songs. She is a tangible product (as tangible as a box with a CD-ROM can be), but she is not a 100% human singer. Miku herself is not a human, but her voice is synthesized by a computer program with the help of a human voice provider to create the initial voicebank. For each sound in the language, voice samples of a human being (in Miku's case, Saki Fujita) are compiled into a voice library. When the producer writes song lyrics using the VOCALOID software, the software locates each sample for the sounds that make up the lyrics and pitch-shifts the sample so that Miku sings the correct note. Then, it runs some proprietary sound algorithms on the notes to make them transition seamlessly so the singing sounds more like a natural human being. I hope this answers your question.