As with many instances in the progression of intellectual history, in order to understand the now, it is important to understand what has come before the now. With specificity to Nietzsche, it is important to recognize the influences of European philosophers, psychologists, scientists, and even musicians upon him (an aside, music was an important aspect of his life). Specifically, such names that come to mind are Schopenhauer, Kant, Hegel, Darwin, and who can forget Wagner! Each of these individuals, and more, produced an impact upon his thinking, whether in his approval or disapproval of their ideas, or even more analytically, a strange, sometimes ironic mixture of both. I highly suggest reading Friedrich Nietzsche by Curtis Cate, who crafts an excellent and truly enlightening biography of the personal and intellectual history of Nietzsche, which has helped enormously in understanding his later works.
I think it somewhat vain for me to try to "correctly" answer how to "properly" read Nietzsche (or any philosopher for that matter), so I will outline how I approach his texts. Remember, this is after having read the biography mentioned above and having gained the reference, context, and knowledge of the milieu and events surrounding the numerous inceptions-and completions-of his many works. Furthermore, the astounding work of Nietzsche scholars (such as Walter Kaufmann) in the form of historical and comparative research has produced conveniently organized footnotes in most texts that I urge you not to ignore. This advice, I think, is also not particular to Nietzsche, but is a generally helpful and advantageous approach to reading philosophy in general. And as a final note, do not forget that the English text is and always will be a translation of the original German; take that as you will.
For larger, continuous works (i.e., not aphorisms):
Read every word, with particular attention to unfamiliar vocabulary. Nietzsche, as a philologist, was particular about his lexical and syntactic choice, I'm sure. Performing a close read of the text will benefit you enormously, particularly when you understand the then contemporary, historical meaning of his words and phrases, for which you will need the aid of footnotes and research. Taking his words merely for granted in the modern definitions and ideations proves inaccurate. Of course, what you are reading is a translation, which I am not qualified to evaluate, but a keen attention to every word, sentence, and paragraph will fully bring you into the experience, and set you up for the next "step".
Note the stylistic, punctuational choices. You will notice that many texts use italicized, parenthesized, or hyphenated text, which to me at any rate, presents a thrilling experience for reading. And this makes sense, given Nietzsche's own opinions on how to write effectively, and I believe such mechanics were present in his original drafts. Feel the words, the phrasing, the tempo, the gravitas, of what you are reading, as it will convey far more emotion and impression than a mere clinical clean read. It should 'disturb' you and make you think, which prepares you for the next "step".
Read the text again, but at a different time. If you have this luxury, try reading the text after some time off, or perhaps in a different mood (one that is still conducive to reading, though). You may be surprised, offended, or confused by what he writes, which dangerously lends to the temptation of dismissing his ideas and style. Understand that he may be speaking ironically, craftily, or earnestly, but all with intent and purpose. He is not an easily philosopher to understand!
Ruminate! This is the most important step, and is not necessarily the last. Think, think, think, about what you have read, and consider the implications of his writing. Nietzsche was extraordinarily productive and crams so much in so little space. Think, at any point in time in your reading or even just in the everyday, about what could have led him to write what you read, and that exactly, and not something else. Think about particular paragraphs, sentences, even words, but without forgetting an understanding of the overarching themes of his message.
As for his aphorisms, given their pithy and brief nature, you need to think long and hard about them, and not cave to the temptation of appropriating them out of context. Furthermore, it is beneficial to seek the expertise of Nietzsche researchers, who can better provide the context and clarity of how and why he wrote with an affinity for aphorisms. I could write on and on, but I hope this is a good modus operandi for approaching his fantastic works. Best of luck!
