I have not read all the posts on this thread, so I apologize if these have all been mentioned somewhere.
1. Assassination would be the most obvious solution. However, if your character has no well known claim to their throne, then unless it is the kind of kingdom where the strongest rule (if you kill the king then you rule), then assassinating the king would have no effect. The next in the line of succession would take over.
2. Causing a coup. Being accomplished by convincing the people of the kingdom of their rulers corruption, convincing the people that your character is the rightful ruler, etc.
3. Convincing a foreign power to back your character's bid for the kingdom's throne. Maybe your character can convince said foreign power that your character is the rightful ruler and that they are doing the "just" thing. Or maybe your character can tell the foreign country that it is in their best interest to back him (your character will grant them some of the local resources of their kingdom, convince them that the current king of the kingdom in question is planning to attack, etc). While this "foreign power" is usually thought of as another kingdom, it doesn't have to be. For instance, it could be a dragon, trans-dimensional, space witch, etc. It can really be whatever you want, whatever fits best with your story.
4. Convincing the king of your character's right to be in the line of succession. Perhaps you really are their son, perhaps you poisoned the king, causing his mind to unravel, the point is that he believes you are a rightful heir and therefore, it is now law. Now all that needs to happen is for the king to meet an untimely end, allowing your character to ascend to the throne.