So I noticed that the hiragana character: は can be pronounced as 'ha' or 'wa' depending on the sentence.

eg. 私は。。。--> when said, it becomes 'watashi wa'

my question is, when is it pronounced 'ha' and when is it pronounced 'wa'?

The character is pronounced 'wa' when it is the connective (between words to separate the subject) in the sentence. If the character is part of a word, then it is pronounced 'ha'.

That's why 私は。。。 is pronounced 'watashi wa...'

While おはよう is pronounced 'ohayou'

That's the basics of it. Don't know if there are more complicated rules about the use of it.

ha is only pronounced as wa when its a particle coming after a subject such as like s0l1dsn8k3 has said it connects but according to my Japanese teacher the meaning is "is". There are many particles in the Japanese Language so you have to look out for what you actually say.