It seems possible.

Yes, the PC is a Dragonborn, just like the Septims, but no I do not believe that s/he will become the Emperor.

It would be a pointless move by Bethesda if they so chose to implement this in-game, or write it into the Lore, and it would be a pointless addition to Skyrim because it would not serve a purpose (Like being a Guild head in Oblivion/Skyrim).

This has nothing to do with dawnguard

Apologies posted into the wrong thread.

I don't know but I have a feeling that the Empire is weakening and possibly near its end/slumber. The Aldmeri dominion wins with the civil war, and with an even more segregated empire(Elsweyr/Summerset Isle - Orsinium/Cyrodiil/Morrowind - Skyrim/Hammerfell) I believe we'll see an era of Elven dictatorship... Id wait to see who High Rock and Argonia is going to support, and if Akavir is going to meddle in the war.

High Rock is an Imperial province, Black Marsh has just once fought an offensive war against a weakened near destroyed enemy. Even if they were to join the war they would likely not be that big a deal, more important however is the fact that they probably won't join a war. They are quite isolated from the rest of the world and they like it like that.

Anyway on to the original question, you won't become Emperor because you are a far to well known person.

I find it unlikely for a couple of reasons.

The Race of the Dragonborn will be left a mystery, much like the Race of the CoC and the Nerevarine was never mentioned after the event. This makes it very hard to narrow it down to a potential bloodline that could follow. They could go with the standard Imperial royal family, but that would, fromm what we know about mating in TES, either limit the gender to male and race to Elven or Human. Of cours this is a fairly broad spectre, but it is only one of the option.
If we imagine the Dovahkiin as female we would have to limit said Hero's race to the race of their decendants, which impose a much more strict and solid information. To have solid information about who and what the Heroes of the main ES games were would be breaking a tradition going back to the very first game.
The race issue will also almost certainly limit the race possibilities a canon Dovahkiin to Nord, Imperial and Breton seeing as they are the ones who traditionally have ruled the Empires.

From a lore standpoint it seems very unlikely that it can happen. Shivering Isles spoiler:
The second argument against it is from a gameplay perspective. As Emperor there would have to be a major overhaul in the machanics in the game since they are not designed to be used in a game were you govern an Empire or Kingdom. If you ebcome Emperor you would have to have a near constant presense in the Imperial Palace and you would be largely limited as to where you could go and who and how you could interact with people.

Questing and exploring would be elements in the game that would fall out, and with the general game design the ES series follows this would kill the game for most people.
Another SI spoiler:
That is just my two cents.

The empire led by a khajiit empress? No, i don't see it happening

Nor by a orc barbarian, altmer mage, dunmer assassin, etc, etc...

"And the Emperor/Empress of unknown origin, gender, race or name, known only as Dovahkiin, ruled the Empire in prosperity. Nothing else is known about this mysterious hero."

I don't see it happening. Emperors and Kings don't go adventuring and dungeon crawling, they sit in stuffy throne rooms handling politics.