Well, first up, I'm assuming that you're using handbrake correctly with the settings and such for the proper levels of compression for the filetype you're converting it to. If not, trying adjusting the compression settings to see if it reduces the file size down. Any time you're working with video there's always a tradeoff between file size and compression levels, so you have to find that happy medium where the quality/filesize ratio is acceptable to you.
As for the computer getting slow, handbrake is a good multi-threaded program which will utilize all cores available to get the job done. Depending on your laptop specs - unless you have a quad core hyperthreaded i7, it's going to slow things down. You can adjust the rendering settings to only utilize as many cores as you want if I'm not mistaken to keep your laptop running a bit smoother but it will increase render time.
To be honest, using a laptop to render videos (unless it's a high-powered workstation designed for it) will result in slowdowns because a laptop is geared towards lighter usage while a full desktop machine is better suited towards rendering and a decent i5 quad will rip through videos at a vastly increased speed. Plus, with a desktop you can have a dedicated OS drive and a big, dedicated drive for processing videos.
tldr; It's a laptop, not really designed for video work. Build a small desktop for dedicated video processing.