Warning: I'm old-fashioned on Catullus. Not that I believe in him as a poet of unmediated passion and romance, but that I am most interested in his place in the Roman poetic tradition and social context. The two books that I find most useful still are:

D. O. Ross, Style and Tradition in Catullus (Cambridge MA, 1969)

T. P. Wiseman, Catullan Questions (Leicester, 1969).

Both of them have written more, and more recently. For Ross, see the first chapter of Backgrounds to Augustan Poetry (Cambridge 1975) and for Wiseman, a whole string of books, notably Cinna the Poet and Catullus and his World.

The basic questions (to me) remain:

What is the significance of the arrangement of the poems? (traditionally: polymetrics, 1-60; longer poems, 61-68; epigrams, 69-116) Were they arranged by Catullus, or by an editor?

Can one establish an internal chronology of the Lesbia poems? Note Wiseman's important demonstration that no poem can be dated (in terms of references to external events or people) before 56 or after 54.

What is the relationship among particular poems (e.g. on Lesbia) in the three sections of the collection? In particular, what language (images, metaphors, word-formations) is used in the polymetrics and the epigrams? If you read each set of poems without knowing the other set, would you describe the relationship in the same way?

What is the relationship between the mythic narrative of poem 64 and the erotic poetry?

Poem 64 itself has a raft of problems, and is extraordinarily difficult. What is the relationship between the "outer" and "inner" narratives (Peleus/Thetis and Theseus/Ariadne)? How does C. characterize the heroic age, and is the poem optimistic or pessimistic (to put it crudely; the same question, of course, applies to the Aeneid)? How does the poem work as narrative, and how is it related to the Alexandrian and earlier Roman traditions? Before reading poem 64, read again the fragments of Ennius' Medea. How and why does Catullus use Ennius?

The bibliography on this poem alone is immense; you can get a start by looking at the intelligent discussion in chapter 2 of S. Hinds, Allusion and Intertext (Cambridge, 1998).