How can we get rid of monarchy in Britain?

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.