How do you catch a pokemon in a trainer battle?

after entering the code go to whatever trainer you want to battle and try to catch there Pokemon

No it isn't possible through normal gameplay in any game of the series unless you count the Snag Machine in Pokemon XD and Colluseum.

Can't.

ob. no but you can migrate from diamond and pearl.

In Pokemon colociem you can catch Pokemon like charizard,pikachu,pidgey,not realy Pokemon in diamond and pearl or plattinum or emerald, sappire, and ruby. plus you will find some from Pokemon red...