Hebrew (5-6): "Turn, LORD; deliver my life [nephesh]. Help me because of your faithfulness [chesed]. For in death there is no remembrance of you, in Sheol who will praise you?" [the last phrase is really neat--who will "throw" to you, an idiom for giving God thanks and praise]
LXX (5-6): "Turn, Lord, rescue my life [psyche]. Save me because of your mercy. For in death there is no one who remembers you, and in Hades who will confess you?"
It seems impossible to know the original context, date, and other particulars of this psalm. Many commentators think of it as a "psalm of sickness" (although it is one of seven penitential psalms of Christian liturgy). Yet to me the psalm seems more about fear because of enemies than sickness.
While some argue for a dependence of the psalm on Jeremiah, others argue that the turns of phrase are traditional enough formulations that no literary dependence can be proved. The author of the psalm clearly connects his (likely a he) sickness with God's anger.
By the way, I do not personally believe we have enough evidence to know whether David himself authored a psalm such as this one. The titles came later than the composition of the psalms themselves, and the expression they use ("mizmor ledavid") is itself a bit odd. I'm sure it can be translated something like "a psalm [attributed] to David." But it does not seem to me the most natural way to say a psalm of David.
In the end, it does not seem that we can date this psalm with any certainty to before or after the exile or or link it to the temple or determine any of these things we would like to know. Its presence in Book 1 of the Psalms may indicate an earlier rather than later date and in principle I am not opposed to it coming from David's mouth.
The most straightforward way to take this verse is that the psalmist does not believe that the dead are conscious of their shadowy existence. The dead do not "remember" God, and they do not "throw praise" His way. So the author asks God to rescue him from death, and affirms in faith that he will (6:9).