You are allowed to install your key on multiple computers as long as only YOU are using the software. The License Agreement shown when you install Newsleecher says "You agree not to share your Key with anyone. It is your personal Key and may not be used by other individuals. You may however use the Key on as many computers as you wish."
Be aware that most news providers do not allow connections from different IP addresses at the same time. So although there isn't a limitation with Newsleecher, you might not be able to download from multiple locations at the same time.