Unblocking
The well-known method is to pour boiling water and add some detergent. It does work but it may need a few pots of boiling water. And it works well if the toilet is slow rather than totally blocked.
When your toilet is totally blocked, the likely reason is too much toilet paper. In that case, you can pour like 50 gm of percarbonate into the toilet after putting hot water in it and leave overnight or for several hours. Percarbonate is the major component in OxiClean powders. So instead of buying the fillers in it, I brought the pure thing.
The hydrogen peroxide in the percarbonate seems to be the only thing that can "dissolve" toilet paper. It oxidizes (burns) toilet paper slowly and doesn't turn them into other solids.
Strong acids such as sulphuric acid wouldn't work. It just turns everything into black carbon that doesn't dissolve. It's a big mess. Washing soda is a strong alkaline (base) that is very good at dissolving organic matters like human solid waste. But it wouldn't work on unblocking toilets, because it doesn't attack toilet papers.
You can also use hydrogen peroxide for first aid. You can see that your skin is burning slowly with bubbles coming out. But the washing soda in the percarbonate is also good for cleaning the toilet.
So after unblocking the toilet you also have a very clean toilet.
Cleaning below the waterline
It's easy because you can soak the toilet with whatever minerals you desire for any time you want. Strong alkaline like washing soda attacks organic matters, which consist of mostly acids, like DNA, amino acids, etc. But I'm not an organic chemist. If you touch the washing soda solution, you feel slippery because your skin is dissolving. Organic dirt are dark, like human waste, and mold.
A mild acid can dissolve the dissolved solids in water like calcium and iron compounds that are deposited on the surface. This inorganic dirt are white, or colored stains from yellow to orange.
And hydrogen peroxide is the super-agent that accelerates either process because it doesn't react with alkaline nor acidic.
Since percarbonate contains hydrogen peroxide and washing soda, you can soak the toilet alternatively with 50g of percarbonate and a cup of vinegar. I prefer to use citric acid powder because it's easier to store and it's odorless. About 50 g of it will start to have effects.
You will have a very white toilet unless the surface is damaged. And hairline cracks will disappear.
Cleaning above the waterline
This is the tricky bit because any cleaning liquid just runoff. I use 100g of baking soda (not washing soda) and add 50g of hydrogen peroxide to it to make a paste. The paste can stay on the toilet walls and under the rim.
To apply the paste I use a wash bottle. 
Take out the inner tube. Invert the bottle. And you have a precise applicator that can produce a jet that reaches the underside of the rim.
After the paste dries, I remove it with 20% citric acid solution, 20g of citric acid powder per 100g of water. I spray the paste with the acid solution in a wash bottle. It's the principle of using alkaline and acid alternatively to remove organic and inorganic dirt. But somehow the mixing of baking soda and acid seems to produce extra cleaning power. It may be the gas that produces some physical movements. Maybe there are some transient chemical reactions.
Touching up
If there are visible dirts remaining, you can repeat the cleaning process. But very likely the speck of dirt can be dislodged by a stream of water from a wash bottle.
Now we are going to talk about some touching if you consider it that way.
If you have a speck of dirt that doesn't seem to move. A quick way is to rub the surface with an ice stick embedded with toilet paper in it. You can find ice stick makers for making ice in the freezer to put in water bottles. Before leaving the water to freeze, roll up a square of toilet paper to put it into the water, leaving a little bit out of the water. When you are done cleaning, throw it into the toilet; it's just water and toilet paper. If you can dispose of it in the toilet, it's as good as it never touched anything.
There are other things that you can use as a small disposable brush. A dental cleaning sponge is one. It's like a small lollipop but the end is a small sponge to wipe clean teeth. Or long q-tips.
If you still want to use a toilet brush, it's a lot quicker. If you follow some of the procedures you can use the brush or plungers when the toilet is almost clean. I can't stand using them when the toilet is a big mess.
