The title Hardrada meant hard ruler which should give you an idea of why Harald deserved to be king of Norway. However, you are mistaken in thinking that he was Norman. He was Norse or a northman. Perhaps you are confusing him with William of Normandy who defeated the English forces at the Battle of Hastings mere days after Harold Godwinson, the English king, heavily defeated Harald's army at the Battle of Stamford Bridge where Harald's claim to the English throne ended with his death.
Now, you ask why Harald deserved to be king. I think you should be asking why he wanted to be king instead. I think the answer you want is the one I have copied here.
Harald Hardrada...
"had come to England with the idea of claiming the English Throne as his own, basing this claim on a supposed agreement between Magnus and Harthacanute whereby if either died without heir, the other would inherit both England and Norway. Instead, he met his end."
Now you need to find out who Magnus and Harthacanute were if you are to answer your question fully. The link below will take you to the article from which I have taken the above quote.
I would suggest that for future reference, rather than ask a question like this one and then wait for someone to answer it, use Google. You will usually find exactly what you want and in a very short time too. Finding this information for you took me all of two minuted. Anyway, it's far more fun researching and finding out the answers for yourself. So off you go and find out who Magnus and Harthacanute were. I hope this helps.
