In modern Britain, the monarchy remains in place because of laws passed by Parliament. That means that "getting rid of" the monarchy today could simply be done by changing the laws. There is a movement within the UK to do this, though it has not gotten much traction with the public.
Since the British monarchy is subject to law, it could be abolished by act of Parliament. People who want that to happen should engage in the kinds of actions that are usually taken to try to change laws. As you can see from this link, groups that want to abolish the monarchy do things like trying to change public opinion on the issue and persuade members of Parliament to act to end the monarchy.
If you want to end the monarchy, then, your best course of action is to engage in politics. You will need to promote your idea among the public and then use public opinion to pressure Parliament for change.