What is the maximum weight could 26 1 3/8 tire and tube could withstand? I bought my first bike few months ago, and it is old Raleigh 26" 1 -3/8 size. 
Every time I go with this bike through some bumps the rear tire and inner tube get punctured with large hole. 

It is written in the tire that the maximum weight is 70Kg and I am 80Kg. 

Is it the quality of the inner tube and tire makes it vulnerable to puncture? (we only have some cheap chinese tires and tubes in my country)

Or the size will not fit with my weight?
Or I can find a 26" tire and tube that could hold my weight?
You are not putting 80 kg of weight on either tire; you're putting 80 kg + bike weight (say, 15 kg) on both tires combined, probably about 60% on the back, 40% on the front. So, about 57 kg on the back. In any case, I doubt very much that your weight is making you more susceptible to flats.
The quality of the inner tube doesn't affect puncture resistance very much. The tire does determine puncture resistance. Some tires are engineered for light weight, some for flat resistance, some to meet a price point.
One important cause of flats is under-inflation, which can happen with any kind of tire or tube. This permits the tire to bottom out against the rim when you hit a bump, causing what's called a "snakebite flat." So a first step towards preventing flats is ensuring adequate inflation. Here's an inflation calculator.
The other main cause of flats, as you might imagine, is road debris. There's not a lot we can do about that other than try to avoid it. If I unavoidably ride through a patch of broken glass, I'll try to wipe down my tire with my (gloved) hand immediately after, but realistically, the damage is probably already done. If you think this is a likely culprit, you might try tire liners, which fit between the tire and inner tube and provide an extra layer of protection.