I have Toshiba TECRA A8. It has "battery drinking problem". It does not give battery back up not more than 5-10 minutes. how to check if the problem is with the battery or the computer as i dont have other battery to check.

the same laptop has another problem. The display is on but it is too dim that you could hardly see anything. But sometime it works fine for few hrs and its very rare. Does it be solved if i open it and clean it or see if any loose connections are there inside. The display is also slightly missplaced. May be it is due some loose connections so is it a good idea to open it and see it and clean it myself, i want to give it a try on myself because i am familiar with mobiles cleanning and repairing it with sometime and a bit with PCs as well.

Both problems are undoubtably related. Could be battery, charger or both.
Both could have gone bad and eventually do, unfortunately, (short of an obviously damaged power cable) neither is user repairable.
Cleaning the internal battery terminals is iffy but MIGHT work, then again, it MIGHT NOT.
Best to start with a new charger, its bound to be the cheaper of the two. If that doesn't help, you'll know what to spend your money on next.