The other answers are good for explaining the grammar. However, I would add that an important part of translating any text is remembering the context in which the passage was written. (I realize that the other answerers probably subscribe to this platitude as well.)
So, let's look at the opening lines of the psalm (taken from Douay-Rheims):
[2.] Benedixisti, Domine, terram tuam; avertisti captivitatem Jacob.
[3.] Remisisti iniquitatem plebis tuae, operuisti omnia peccata eorum.
[4.] Mitigasti omnem iram tuam, avertisti ab ira indignationis tuae.
[5.] Converte nos, Deus salutaris noster, et averte iram tuam a nobis.
[6.] Numquid in aeternum irasceris nobis? aut extendes iram tuam a generatione in generationem?
[7.] Deus, tu conversus vivificabis nos, et plebs tua laetabitur in te.
[2.] Lord, thou hast blessed thy land: thou hast turned away the captivity of Jacob.
[3.] Thou hast forgiven the iniquity of thy people: thou hast covered all their sins.
[4.] Thou hast mitigated all thy anger: thou hast turned away from the wrath of thy indignation.
[5.] Convert us, O God our saviour: and turn off thy anger from us.
[6.] Wilt thou be angry with us for ever: or wilt thou extend thy wrath from generation to generation?
[7.] Thou wilt turn, O God, and bring us to life: and thy people shall rejoice in thee.
As you can see, there's a lot of 'turn' language in this passage. So, there is no 'again' in the strict sense, but Ps. 84:5 especially explains why the translator did not feel it was out of place to add it in.
The 'future' tense here, 'wilt turn' is explained by virtue of the fact that it is controlled by the tense of the main verb (as others have mentioned).
What will he have turned (himself) from? His anger, of course.
Although I don't think we need advanced theology or literary degrees to come to that conclusion, it might be worth looking at the so-called Glossa ordinaria (and I'm simplifying what that refers to) on that passage.
In this image (taken from the a scan of an edition from Basel, 1498, you can see how several possibilities were aired:
The large text in the centre is the Psalm and some small interlinear glosses. In the left margin, in red, you can see :
Deus conuertens. quasi sicut uiuificatio est a te: ita et conuersatio. h Deus tu conuer. Casus. Non extendens. Set deus conuertens. Prius dat uotum conuersitionis [sic]: post ad uitam ducit. Uel: Deus tu conuersus. ab ira in aduentum filii.
God converting. As if: just as vivification is from you, so too is conversion. h Deus tu conversus: Overview/summary. Not extending; but God converting. Earlier he gives a vow of conversion; after, he leads to life. Or: Deus tu conversus: from anger at the advent of his son.
(The h is a the (late!) medieval equivalent of a footnote here, to help readers track where to find the commentary on the relevant passage. Thus, I'm actually starting a little before the lemma we are interested in.)
On the right side, there is the more succinct:
Deus tu conuersus. ad nos per tuam benedictam incarnationem.
Deus tu conversus: toward us, by means of your blessed incarnation.
And in this impage from a subsequent page
you can see in red that the vivification will be due to his augmenting of grace in us:
The line reads:
Deus tu conuersus uiuificabis nos. gratiam in nobis augmentando
Deus tu conversus vivificabis nos: by increasing the grace in us.