In onee-chan the "e" is pronounced like the "e" in "egg." Since it is a double "e" the sound is held twice as long but the pronunciation doesn't change.
In onii-chan the "i" is pronounced like the "ee" in "keep". Again the double "ii" is held twice as long without a change in pronunciation.
On an unrelated note, attaching "sama" the end of someone's name is a polite way to refer to someone else, but shouldn't be used when referring to oneself. The same applies for attaching "san" to names.