Non-modal approaches to (de dicto) modality aim to give an analysis of modal concepts where the analysans (the analysing concept) purportedly doesn't involve any modal notions.
Lewis's realism is an example of such an approach. To see this note that Lewis's analysis of de dicto modalities involves two claims, the first one being an old Leibnizian idea.
It is necessary that p iff p is true in all possible worlds.
The second claim answers the question what possible worlds are. Very roughly:
A possible world is a maximal spatio-temporally interrelated whole.
Some comments: A whole is something that has parts. A whole is spatio-temporally interrelated (sti), if each of its parts stands in some spatio-temporal relation to some of its parts. A sti whole is maximal, if any entities that stand in some spatio-temporal relation are parts of the whole.
Now, Lewis claims that none of these defined concepts is modal; so, neither is Lewis's concept of possible worlds. Since necessity is defined in terms of truth in possible worlds necessity, according to Lewis, has a non-modal analysis.
For a good overview on theories of modality see Ted Sider's 'Reductive Theories of Modality':
For Lewis's views on modality there is no better source than his monograph 'On the Plurality of Worlds'. If you want to know more about possible worlds read John Divers's book 'Possible Worlds'.