The first sentence doesn't work at all. I would amend the second as follows:
My academic background is excellent (note two "l"s in excellent): I was awarded the _______ grant to support my research in __________ (or I received the ______ prize for my research in __________).
What I'm saying is that you need to put the supporting evidence for your "excellent academic background" immediately after your claim.
(One's GPA is usually just indicated on the C.V., without bragging about it separately in a cover letter. If you need to write it separately. I would make it the last sentence of your paragraph, after the more important information about your rewards and your research.)
I have and excellent academic background is the only one that is correct in this case because the verb "have" is in the present tense but you are speaking of the past. If you use the other it must be in the past tense and you would say: I have had an excellent academic career.
Yes, that is gramatically correct. However, I've spent many years in academia, and I can tell you that no one would ever utter that sentence.
If a person has left academia, they might say, I have had a wide-ranging academic career, or I have had a satisfying academic career, or I've had a fascinating academic career. But never 'excellent': it's just too immodest!