I'm thinking about making an Imperial warrior, who will eventually become a Civil War hero, and of course The Mighty Dragonborn!

Is it going against the lore though? I'm not quite sure, so an explanation would be helpful!

Not at all, any race could be Dragonborn. Though the figure is seen as nord, but that does not mean you can't be somebody else.

I understand that. All races CAN be Dragonborn, but for some races it just sounds weird.. Khajiit and Argonian for example.

Yes.

Both Saint Alessia and Reman Cyrodiil were Dragonborn and Imperial. Well, Alessia probably not so much but she was of a tribe that could be considered a more direct ancestor to the Modern Imperial.

Imperials are Nords, so there's really no point in differentiating between the two.

The real crux is I don't think the Nords of Skyrim would accept any other race as being their culture hero. If you were an Altmer Dragonborn I think you'd have to Shout everyone to death to actually prove it to people.

"Dragonborn? You? A filthy Elf?"
"FUS RO DAH!"
"You don't expect me to..."
"FUUUS ROOO DAAAAAHHHHHH!!!"
"Ok, I see your point."

The Dragonborn are blessed by Akatosh. He bestowed the 'gift' on mortalkind, not on any particular race(s).

I'm not going to argue with him.

Reman Cyrodiil was DB and an Imperial (Hence the name)