What did British forces do during their invasion of the United States?

There was no invasion during the War of 1812.

British strategy was to force the US into a negotiated peace through crippling its economy, and one way to do this was through a series of destructive raids on economic targets.

British standing orders to commanders were "to destroy and lay waste such towns and districts along the coast as you may find assailable" (ie undefended) - hardly orders to get ready for and carry out an invasion. British forces retreated after burning Washington D. C. in August 1814 when there were no US forces in the area threatening them, again hardly the action of an invasion force, which would be expected to maintain possession of the enemy capital once they had occupied it.

So, British incursions into US territory were designed to damage economic targets rather than to permanently occupy areas with a long term view to eventual conquest.

captured and burned Washington, D. C.