This question assumes a few facts that could very well be untrue:
Jon Snow is the child of Lyanna Stark and Rhaegar Targaryen and he wishes to claim the Iron Throne.
Given these facts, he would have a valid claim by blood, but this claim would be hard to prove since Ned Stark is dead.
How could he assert his claim on the Iron Throne?
This question isn't a duplicate since I'm not asking how valid his claim is. I'm asking how he could assert his (possible) claim on the iron throne.
The scene we see in Season 6 Episode 3, Oathbreaker where Bran is having his flashback.
This scene is known as the Tower of Joy. This was Ned's attempt to rescue Lyanna. Howland Reed was the only other survivor of the encounter. He is still alive at this point and is the Lord of Greywater Watch. Howland will be able to support Jon's claim.
It is also possbile that Jon was legitimized by Robb and may have claim to Winterfell.
In A Storm of Swords, Robb and Catelyn have the following conversation:
“Mother.” There was a sharpness in Robb’s tone. “You forget. My father had four sons.”
She had not forgotten; she had not wanted to look at it, yet there it was. “A Snow is not a Stark.”
“Jon’s more a Stark than some lordlings from the Vale who have never so much as set eyes on Winterfell.”
“Jon is a brother of the Night’s Watch, sworn to take no wife and hold no lands. Those who take the black serve for life.”
“So do the knights of the Kingsguard. That did not stop the Lannisters from stripping the white cloaks from Ser Barristan Selmy and Ser Boros Blount when they had no more use for them. If I send the Watch a hundred men in Jon’s place, I’ll wager they find some way to release him from his vows.”
He is set on this. Catelyn knew how stubborn her son could be. “A bastard cannot inherit.”
“Not unless he’s legitimized by a royal decree,” said Robb. “There is more precedent for that than for releasing a Sworn Brother from his oath.”
Robb then has a council with his lords as follows:
He picked up a sheet of parchment. “One more matter. Lord Balon has left chaos in his wake, we hope. I would not do the same. Yet I have no son as yet, my brothers Bran and Rickon are dead, and my sister is wed to a Lannister. I’ve thought long and hard about who might follow me. I command you now as my true and loyal lords to fix your seals to this document as witnesses to my decision.”
It's not clear what has happened to that sheet of parchment. Robb then sent Galbart Glover and Maege Mormont to Howland Reed at Greywater Watch, but the written messages they carried were false, in case they were captured.
Daenerys will meet Jon when she comes west, as foretold in the prophecy she received in the house of the Undying of Qarth
A blue flower grew from a chink in a wall of ice, and filled the air with sweetness. .. mother of dragons, bride of fire...
Daenerys is the rightful heir to the throne, her dragons have been foreshadowed (for example in the scene with Brown Ben Plumm) as having the ability to "smell" Targaryen blood, and likely she will come with a huge army at her back and just conquer Westeros. When she finds Jon, he claims a dragon, they marry, done deal. :) That is to say, Jon will take the throne with Dany's help.
That is how I've always seen it happening in the books, at least. In the tv-show, it may be different.
I believe claiming the throne in Westeros is no different than claiming a throne in the medieval Europe. Anyone can claim throne but if you want people to take you seriously, you need to have three things:
Legitimacy: If the R+L=J is true, Jon has a legitimate claim to the throne. We do not even need Daenerys to legitimize this part.
Capability: Jon demonstrated his capability as the Lord Commander of the Night's Watch. Because of his successful ruling, many wildlings now follow him.
Power: Jon currently possesses little power to be taken serious. But, if/when he unites the North, if Iron Bank supports his cause as they did with Stannis, and if Daenerys joins his cause, he might then attract attention of the other houses.
Note that Daenerys has all three of these right now.
