So I arrived near that big ship and proceeded to climb to enter the town of Solitude. Since it was dark, I waited.
When it was morning after my wait, there were these 3 large Nords who attacked me. They killed me instantly when I tried fighting them.
I don't know who they are, I hadn't done anything wrong, and I am not even part of any faction.

They're not bandits...
On my second attempt I ran all the way up to Solitude (with them at my heels!) and went to the place where there are a lot of guards, thinking they'd take them up for me as they'd do with bandits...

But no. The guards ignored them and they kept following me. After I entered a couple of houses and waited some more, I exited back to Solitude outside. Still scared stiff if they'd find me!

Any clue on who these people are and why they want to kill me? They're not guards, they look like bandits, and their boss is wearing a horned helmet.

I'm not sure. Sounds interesting though.

I wish I Knew! Sounds crazy!

If they're not bandits, what are they? It should still tell you their identifier.

And since this occurred after waiting, are you a vampire?

I had something similar happen to me, they IDed as something like Hired Thugs, Turned out they were sent after me by someone i'd stolen from earlier. Very tough since it happened to me early in the game.