It should be /ˌhɑɪpoʊˈkaɪmənɒn/ “high-poe-KIME-uh-non".
According to Wikipedia, this word is taken from Greek ὑποκείμενον (it's actually a letter-by-letter transliteration, aside from the initial “h” which represents the Greek rough breathing mark). The pronunciation of the consonants is unproblematic.
The stress is clearly on the third-to-last syllable, as all the relevant languages involved agree on this (Greek, Latin and English). The prefix “hypo-” already has an established pronunciation in English, which I would advise using (/ˌhaɪpoʊ/, or “HIGH-poe”).
The pronunciation of the last syllable is a bit harder to determine, as the Ancient Greek neuter suffix -on is usually converted to the equivalent Latinate suffix -um in English words. (For example: bacterium, from Greek βακτήριον baktērion. Cerberus points out in a comment that the fully Latinized version of "hypokeimenon" would be hypocimenum.) However, a good analogy for the vowel sound can be found in the word automaton, from Greek αὐτόματον automaton. Some people pronounce it as /ən/ “un, while others pronounce it as ”/ɒn/ “on.”
The second-to-last syllable is unstressed, so will likely simply have a schwa sound in it, /ə/, “uh” or “ih.”
What remains to be determined is the pronunciation of the vowel in the stressed syllable. In general, the digraph “ei” in Greek-derived words, which corresponds to the Greek digraph εἰ, represents a “long i” sound (it is sometimes even interchangeable in the spelling with an “i"). Traditionally, English speakers have pronounced this as the English “long i” sound, /aɪ/, as in the words Deimos, meiosis, eidolon, or irony (from Greek εἰρωνεία eironeia). Some people might try to use a pronunciation closer to the Modern Greek, and instead use /iː/ (an “ee” sound), but this is not as common a pronunciation for this spelling pattern (although it does occur in some proper nouns, such as Cassiopeia).
Although the spelling of the stressed vowel suggests an /eɪ/ “ay” sound to the naive eye, this would be an irregular pronunciation, not in accordance with either the traditional English pronunciation or the modern Greek pronunciation. I would therefore recommend against it. However, there is one argument that could be made for it: it is closest to an early stage of what is believed to have been the Ancient Greek pronunciation of the ει in ὑποκείμενον. There is a variety of evidence that the Greek digraph ει initially represented a diphthong along the lines of /ei̯/, which later developed in Greek to a monophthong /eː/ (and in Modern Greek /i/). I think someone who used this pronunciation would be more likely to use the spelling hupokeimenon, since /u/ is likewise believed to be the earliest vowel quality for the sound represented by the letter υ in Greek (early on in most varieties of Greek, it developed to a front rounded vowel /y/ (like German ü), which lost its rounding to become Modern Greek /i/). Personally, I don't see any reason to try to approximate the Ancient Greek pronunciation of the diphthong in this particular word, since modern English pronunciation has many other significant and irremediable differences from Ancient Greek (a "proper" Ancient Greek pronunciation would use pitch accent, but English speakers naturally use stress accent, and an Ancient Greek speaker would use an unaspirated /k/, while an English speaker naturally aspirates /k/ at the start of a syllable).
So: the overall pronunciation would be /ˌhɑɪpoʊˈkaɪmənɒn/ “high-poe-KIME-uh-non”, or if you wish, /ˌhɑɪpoʊˈkiːmənɒn/ “high-poe-KEEM-uh-non.”