Wordorigins.org

Ghost is of Germanic origin. It dates back to the Old English period and the word gast. The general sense of the word is a spirit or soul and it is used in various specific senses.

The earliest attested context is that of the soul of a dead person. From Bede’s Death-song, c.735:

Huaet his gastae godaes aeththa yflaes aefter deothdaege doemid uueorthae.
(Lo! His ghost, of good or of evil, after death-day becomes judged.)

The same sense, but in the form of giving up the ghost, appears c.900 in an untitled manuscript. Agifan (3rd person past tense = ageaf) is an Old English verb meaning to give up, to give back, to restore. So it modern translation we say give up the ghost meaning to die:

Se casere hio heht gemartyria(n) & God wuldriende heo ageaf hire gast.
(The emperor ordered she be martyred & glorifying God, she gave up her ghost.)

We can see the modern form of the phrase in Wyclif’s 1388 translation of Matthew, a passage about the crucifixion:

Jhesus eftsoone criede with a greet voyce and gaf vp the goost.
(Jesus again cried with a great voice and gave up the ghost.)

Ghost is also used to mean the spirit of God. This sense survives today in the phrase the Holy Ghost. The Vespasian Psalter, c.825, records the following:

Hwider gongu ic from gaste ðinum.
(Whither go I from your ghost)

Finally, the modern sense of the spirit of a dead person that roams the earth doesn’t appear until the Middle English period. From Chaucer’s The Legend of Good Women, in The Legend of Dido, lines 1294-96, c.1385:

This night my fadres gost Hath in my sleep so sore me tormented.

(Source: Oxford English Dictionary, 2nd Edition)