Getting Biblical in Daily Life
Jewish Perspective
Like Moses, the Baal Shem Tov, Einstein, and Bob Dylan, David is one of the immortal icons of Judaism and Jewish identity. Orthodox Jews even continue to believe that the Messiah will come out of the House of David. And he's a classic model for the righteous Jewish ruler, in addition to being the poet who traditionally is said to have written the Psalms, and man with the gift of prophecy. (This Messianic belief about David's house helps explain why the Gospels argue that Jesus was descended from King David on Joseph's side.)
In a way, David is a symbol of the unity of the Jewish people—he unites Israel and Judah, wages wars in the service of God, and (in the story) God's love is concentrated on him as an individual in a way that suggests how God's love is concentrated on the Israelites as a people. (This isn't Cuddle-Fest 1000 B.C. however, since God chastises David and the Israelites harshly, too.)
There are numerous legends and commentaries about David that stray outside the Biblical canon. For example, The Talmud tries to explain away David's sins—his adultery with Bathsheba and murder of Uriah—on the grounds that this was simply an opportunity for David to demonstrate how a righteous man can repent and recover from sin.
Jewish legend also elaborates on his encounter with the Philistine giant, Ishbi-benob, who almost crushes David in a wine-press before Abishai kills Ishbi-benob. (Hey—this sounds suspiciously like the end of the first Terminator.)
Catholic Perspective
For Catholics, King David is important in his own right—as God's beloved ruler and a deeply pious man, despite his flaws. But he's also important as a forerunner and ancestor of Jesus (though not a paternal ancestor, since God is supposed to be Jesus' hereditary Dad). The Gospel of Matthew argues intensely for this connection, providing a full Ancestry.com-style genealogy for Jesus (Matt 1:1-17).
Yet, since Joseph isn't Jesus' biological Dad, and passages in Romans and Timothy suggest that Jesus was biologically related, some Catholic scholars suggest that Joseph and Mary were distant cousins, both descended from King David.
St. Augustine, one of the top Catholic authorities of all time, stated that God's promise in 2 Samuel 7:8-16 to have one of David's descendants build God a house, wasn't meant to refer to King Solomon building God a temple. It was meant to refer to Christ and the creation of the Church. Augustine says this needs to be true since God tells David that, "His house shall be faithful, and his kingdom for evermore before me," whereas Solomon's house was "full of strange women worshipping false gods" (source).
The greatest of all Catholic Poets, Dante, also claims to have seen David in Paradise, where David's soul appears in the center of the eye of an eagle comprised of the souls of righteous rulers. Yeah, it's really trippy: "This, that shines / Midmost for pupil, was the same who sang / The Holy Spirit's song…"
Protestant Perspective
The Protestant view isn't so very different from the Catholic view on David. Protestants also accept the Gospel of Matthew's account of Jesus' lineage. They have a very high opinion of him, too, given his status as a beloved leader foreshadowing J.C.
The major Protestant reformer John Calvin wrote, "David is like a mirror, in which God sets before us the continued course of his grace." Calvin implies that in the same way God leads David away from his sins and gradually helps to improve his conduct, so does God for every one he's decided to save.
Matthew Henry, another theologian and Bible commentator, offered some insights on David's sin in committing adultery with Bathsheba: "Adulteries often occasion murders, and one wickedness is sought to be covered by another. The beginnings of sin are much to be dreaded; for who knows where they will end? Can a real believer ever tread this path? […] Though grace be not lost in such an awful case, the assurance and consolation of it must be suspended."
Muslim Perspective
David—or in Arabic, Dawood—is a really important character in the Koran. David isn't just a blessed ruler but a prophet and a messenger (which is true in Judaism, too: see 2 Sam 23:3). This means that he, like the later prophets Jesus and Muhammad, brought a written revelation directly from God (which is interpreted as referring to the Psalms, based on Koran 17:55). According to one hadith tradition, the Prophet Muhammad said, "The most beloved prayer to Allah is that of David and the most beloved fasts to Allah are those of David..." (source).
Islamic tradition tells different stories about David's life, which elaborate on the Biblical narrative. For instance, it's not Nathan who sets David to rights after seducing Bathsheba, but two angels. And David doesn't see Bathsheba bathing on the roof, but when a bird leads him to a lake where Bathsheba is taking a dip.
Also, there are other wondrous and miraculous stories related to David, such as one about a bell David owns which rings magically to indicate the guilty party in legal disputes (source).
Latter Day Saints Perspective
The Latter Day Saints share many of the Christian denominations' views on David's life, but there are a few key differences—mainly that Saints see David in a somewhat more negative light than most. See, the Saints believe that David was a beloved king of God, but that his sins in seducing Bathsheba and killing Uriah caused him to lose his eternal blessings, along with a transgression David committed in taking more wives and concubines without God's permission.
A Mormon Encyclopedia provided by the Brigham Young University website says that David "must await in the spirit prison [a kind of Limbo] the redemption promised to him." He furthermore lost his state of "exaltation" (breaking the seal that would've allowed him to keep his rightly married wives in the afterlife and live in the highest heaven), losing his wives to another—in this case, when Absalom sleeps with his concubines (source).
Baha'i Faith Perspective
The Baha'i Faith believes that David was a great "Manifestation of God", but he is also more of a "follower and promoter" than an "independent prophet." Whereas the great independent prophets, like Jesus and Muhammad and Buddha, brought a message from God without having direct precursors in their areas, figures like David and Solomon followed in the wake of divine traditions that already existed, helping to promote those traditions and make them flourish.
Also, in the same way that Christians claim that Jesus is a descendant of David, Baha'is claim that Bahaullah is a hereditary descendant of David, as well, since they believe that David's descendants intermarried with Sassanian Persian royalty to which Bahaullah's family line can be traced.