First off, why do you want to add teeth to your rear sprocket? Are you looking for more acceleration? The expense of this acceleration is going to be the engine will have to run faster for the same speed as before the new sprocket. Next I doubt that you will actually feel that big of a difference, we're talking about maybe 5%, assuming that your rear sprocket is somewhere around 40 teeth in the first place.
Next, no you probably won't need another chain, the sprocket will have a slightly larger circumference, meaning you will have to retune the chain tightness for a slightly smaller distance between the front and rear sprockets. I would try it on your old chain before you purchase a new one, just to make sure.
However, just as a side note, replacing the chain isn't expensive and is a good thing to do every so often. I would recommend getting an o-ring chain when you do, there are lots listed on ebay for about $50.