If you want to really get your toilet tank clean, then you need to make sure you have the right cleaning products for the job. Vinegar is a great toilet cleaning solution. Not only is it free of chemicals and naturally antibacterial, it's also an acid, so it will remove minor lime and calcium deposits. All you need to do is pour a couple cups of vinegar in your tank and let it sit for an hour or so, then scrub and flush to rinse. But if you don't feel like something is clean unless the fumes of chemicals burn your nose, then bleach is a good one to use. It's a great disinfectant and will get rid of mold, but it won't have any impact on calcium deposits, so you'll need to use a cleaning product containing acid that's safe for use on porcelain.
If you have some caked-on stains that need a little extra elbow grease, you really need to get down inside the tank. If you don't wish to immerse your hands in that water, then you'll need to drain the tank. All you have to do is turn off the water supply, usually somewhere in the lower back of the toilet, and flush the toilet before you start scrubbing. You will likely find that cleaners work a little better and faster when they're not diluted in water; this goes for inside the bowl, too. Just be aware that chemical cleaners can erode the rubber flapper, which will eventually cause leaks. So while you're down there, be sure to check that your flapper is still in good working order.