Critical reading: No doubt other noders will know more about the various technical approaches to this reading style than I do. The following is (in general) a modern technique, and is as well suited to an atheist reader as one who has a spiritual connection with the work. It's not exclusively modern though, and its roots can be traced back to Origen's observation (c. 220 CE) that the Letter to the Hebrews could not (for a variety of reasons) have been written by Paul.
Critical reading benefits from a little external research, but this is not essential by any means. The key is to consider who set down the text in question, and what they tried to say about God and life in doing so. For example, if we read John 2:1-11, the story of Jesus turning water into wine, it appears (perhaps) a little unclear. It's possible, of course, to treat this passage as a simple narrative, like a news report. But that doesn't then tell us anything, except that some people believed Jesus had turned water into wine. But if we look at John's use of language, here and elsewhere, it's clear that he wants us to look at Jesus' life (partly) in terms of the 'signs' he performed. In John's view of Jesus' life, the presence on earth of the Son of Man is naturally attended by miracles, which are not there solely for the benefit of the people healed, or given free wine, or fed, or saved from storms, but also as a direct sign of the presence of the Christ. The story of the wedding at Cana, with its commonplace setting and Jesus' remarkable actions, demonstrates the force of God's power bursting into the lives of those around Jesus and demonstrating to them, and subsequently to the whole world, that Jesus is the Messiah. This view is not nearly as strongly presented in the other gospels, which is part of why they are called 'synoptic' (same view) in contrast to John's. (The narrative content is another major issue.)
Narrative reading: People often make clever 'classical' allusions to various Bible stories which were once well-known, in the days when scripture knowledge was still a curriculum subject, but which now go unread. A narrative reading of scripture is simple; find a section of the Bible which tells a story, and read it from beginning to end. Judges 4 is a good self-contained feminist tale to be starting with. Books which are specially good for this kind of reading include the Gospels and Acts, Genesis, Exodus, Judges, Ruth 1 and 2 Samuel, 1 and 2 Kings, and Jonah. There's no 'deep' objective in this style of reading, except to appreciate the Bible as literature and history. A structure can be developed by reading a chapter a day from a book each of the Old and New Testaments.
Literary reading: As well as the narrative style described above, it is also entirely appropriate to read the text for its compositional content. This approach is somewhat dependent on the translation, and this is one situation where a particularly expressive text has an advantage over a more linguistically precise one. The Authorized Version is ideal. Passages such as Isaiah 40 are justly famous for their rich use of language and vivid imagery. Poetic chapters such as the Psalms and Exodus 15 are also highly appropriate for such an approach. Other passages, like 1 Kings 22, yield up information about history and geography. A biblical atlas can be useful in finding places like Ramoth-Gilead, which is a disputed site at several different times in the Bible. And even sections of the Bible which seem to convey no story and make little sense to the casual reader, such as most of Revelation, can be studied for their poetic thrust and wide allusive range. Thanks to heyoka and SEF for prompting this addition to the WU.