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 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.