The Bible was the most widely known text in Shakespeare’s culture. It was studied by the learned, preached about from pulpits, and quoted constantly by schoolmasters, parents, and other figures of authority. Even illiterate persons who could not read the Bible knew many of its basic stories and teachings because they had heard them referred to throughout their lives or had seen them illustrated in stained glass windows or in other decorations both inside and out of churches. In short, the Bible was a highly pervasive influence in Shakespeare’s culture, and there is no doubt that Shakespeare was absolutely familiar with it and strongly influenced by it.
Naseeb Shaheen, perhaps the greatest scholar of Shakespeare’s use of the Bible, tabulated 1160 Biblical allusions in his magnificent 800-page book Biblical References in Shakespeare’s Plays (Newark: University of Delaware Press, 1999, page 39; by the way, a new edition of this book is due to be published in late 2011). Of course, since allusions are sometimes open to dispute, there can be no final figure, but Shaheen’s tabulation is about as close as we are likely to come to a definitive number of allusions. Direct quotations are another matter; I am not aware that anyone has tabulated a precise number of direct quotations. For anyone unable to consult Shaheen's book, a deliberately briefer overview of the subject is Steven Marx’s Shakespeare and the Bible (Oxford: Oxford University Press, 2000).
There were, of course, a number of translations of the Bible available to Shakespeare, and while many of them resembled one another greatly, some of them were quite distinctive. Shaheen suggests (39) that Shakespeare most often seems to have used the Geneva Bible, which offered a decidedly Protestant view of scripture. Some recent scholars have tried to make a case for Shakespeare as a Catholic author, but the precise nature of his religious affiliations remains open to argument. In any case, he knew the Bible well.