I've looked up pronunciations of her Greek name as well as english an I can only seem to find results for how the word "cleopatra" is pronounced when I am trying to figure out how someone in her time of life would have pronounced her name.
The Ptolomeic rulers of Egypt spoke Greek rather than Egyptian, and the name Cleopatra is Greek in origin (Kleos meaning "fame", patra meaning "father", hence "having a famous/renowned father").
If you read IPA, the pronunciation of her name would have been something like [kle.o.'pa.tra] -- this is not dissimilar to today's standard English pronunciation, except that the first vowel is nearer to that of "clay" rathern than "clee".
Source: Allen, Vox Graeca: The Pronunciation of Classical Greek