If you are taking the Northern Line route, get on at the front of the train, then the exit to the Piccadilly Line will be right where you get off at Lecester Square. Then it's a quick interchange. However, the entrance to the platform at Euston is at the back of the southbound train so either way you will have to walk the length of a platform.
If time isn't an issue you could always take the short walk to Euston Square and wait for a Circle Line train to come along!