The "trema" over the e in this case simply marks that the previous i makes a sylabble. Hence, there are three sylables in this word: Kly-ti-e. Without the trema, there could be two sylables: Kly-tie.
There are no problems with the pronounciation of consonants, I hope.
Spanish vowels, e.g., are very close to Greek vowels.
"Y" (ypsilon) is now pronounced as "ee" (it, see). In classic Greek, "Y" was pronounced as German "fuenf" (five) or French "tu" (thou). This maked Romans to borrow that letter in this shape for that vowel. In pre-classic Greek, "Y" was heard as "oo" (put, good, moon). At those times, Romans that letter with the shapes "U/V". Romans did not distinguish "U" and "V".
"I" (iota) has always been pronounced as "ee" (it, see). In this position without the trema over the next vowel, it would not make a syllable, however.
There are two Greek letters, E (epsilon and H (eta). In ancient times, epsilon was a short e and eta was a long e. Confusingly, both epsilon and eta are spelled with "e" in Roman alphabet. Now, epsilon is pronounced as "e" (bed, any, every) and iota is pronounced as "ee" (it, see).
In this case, however, there is no confusion: the final e is eta and it can be "ee" according to the new Greek pronunciation. If you wish to follow the ancient pronunciation, use something like the German e in "zehn" (10).