The personality of the Holy Spirit is not explicated in the same way as that of God the Father or Jesus Christ. For a discussion of purported biblical evidence that the Holy Spirit is a person (as opposed to being a personal attribute of God and/or the post-resurrected Jesus or an impersonal "force" as various groups [e.g., Socinians; Jehovah's Witnesses]) hold to, Daniel Wallace, "Greek Grammar and the Personality of the Holy Spirit,” Bulletin for Biblical Research 13/1 (2003), pp. 97-125 (online here).
However, while studying the Gospel of John recently in my daily Scripture studies, I think a strong, albeit implicit, case can be made in favour of the Holy Spirit being a person comes from John 6:63:
τὸ πνεῦμά ἐστιν τὸ ζῳοποιοῦν, ἡ σὰρξ οὐκ ὠφελεῖ οὐδέν· τὰ ῥήματα ἃ ἐγὼ λελάληκα ὑμῖν πνεῦμά ἐστιν καὶ ζωή ἐστιν.
It is the spirit that gives life; the flesh is useless. The words that I have spoken to you are spirit and life. (NRSV)
Jesus, by use of a play on words in the clause το πνευμα εστιν το ζωοποιουν ("the spirit is the [entity] giving life" or "the spirit is the life-giving one") is a reference to the Holy Spirit as a person as (i) the definite article precedes "spirit" (το πνευμα), thus denoting the existence of a person and (ii) a second definite article substitutes for a relative pronoun ("who"; "which"); (iii) the spirit performs an action (giving life), whereas a personal attribute or an impersonal force cannot.
The "spirit" without a preceding definite article at the ned of the verse becomes the predicate nominative of "the words" (τὰ ῥήματα) and thus refer to a quality of being. Such appears elsewhere in the Gospel of John, including John 3:6, a text often used to (correctly) support baptismal regeneration:
τὸ γεγεννημένον ἐκ τῆς σαρκὸς σάρξ ἐστιν, καὶ τὸ γεγεννημένον ἐκ τοῦ πνεύματος πνεῦμά ἐστιν.
What is born of the flesh is flesh, and what is born of the Spirit is spirit. (NRSV)
