Before going the letters route, i would suggest getting the updated (2012) edition of
Lovecraft's Library, which lists a sizeable portion of the books he owned (either those which were known to have belonged to him by physical evidence, or which he mentions having owned). It is of course far from complete; we don't have a complete listing of all his books. But this does list slightly over 1000 of them, and also has citations showing where the information about his owning them comes from. It also has a listing (by HPL) of "Weird &c Items in the Library of H. P. Lovecraft", which includes a section on "Mythology, Folklore, & the Occult" -- it is a rather brief list, so any occult books would be easier to track down. (I do know that one of his major sources was Lewis Spence's
An Encyclopedia of Occultism, which can generally be found for very little cost.)
As far as science books, the index would be very helpful on that; and when it comes to his literary influences... hoo, boy! You've got your work cut out for you! There, I would suggest beginning with his essay, "Supernatural Horror in Literature", preferably the annotated version (again, there is a 2012 updated version to include new material which you would probably find helpful), which also has information on each of the major items named (if a writer is simply mentioned, without specific works, you'll find some helpful information in the notes; where specific works are mentioned, there are separate entries for each of these). One thing you ought to look into is Julian Hawthorne's 10-volume "Lock & Key Library", as Lovecraft relied heavily on that for various items.*
I have not read all the more recent volumes of his letters, such as the full Howard/HPL correspondence or that with Derleth, though I have dipped into them at times; but as I recall from earlier readings of the previously published REH/HPL cycle, there was little of the occult library discussed. However, I do know he asked CAS about this issue, and you might want to look up a copy of Smith's
Selected Letters (possibly through interlibrary loan) for some of this. I would also strongly suggest getting a copy of Joshi's
H. P. Lovecraft: The Decline of the West, which deals with many of the scientific and philosophical books which influenced Lovecraft. Perhaps oddly, some of the most influential were not scientific books as such, but science popularizations, such as Hugh Elliott's
Modern Science and Materialism; and there was Haeckel's
The Riddle of the Universe. HPL mentions these in early letters (
Selected Letters I, numbers 76 and 92, for instance), as having quite an influence on him. Also see if you can track down a copy of Joshi's
Index to the Selected Letters of H. P. Lovecraft. It only deals with the original 5-volume Arkham House set, but even this would save you tremendous amounts of time in looking up various comments on writers or specific books.
I hope these suggestions help, at least to get you started....
*Many of these are available online, though not all; just as most of the items mentioned in his essay can be found at the Internet Archive, Project Gutenberg, Authur's Classic Novels, The Literary Gothic, or one of the other literary/fantastic sites.
Here are the links to the two Hippocampus Press volumes:
http://www.hippocampuspress.com/h.p...e-annotated-supernatural-horror-in-literature
http://www.hippocampuspress.com/h.p-lovecraft/about-hp-lovecraft/lovecrafts-library-catalogue
