System unstable for the last month, my daughters laptop has been behaving very oddly. It runs slow while on charge, shuts down for no reason and gets very hot when in use.

Are you using the provided charger, or are you using an after market "Universal" charger?
If you're using a universal charger, it could be charging the battery too hot and the computer just doesn't understand why the battery is putting out more than the system is expecting.
It could also be that the battery is wrong type. The voltage may be the correct, however the Ah may be too high. This being the case, the computer doesn't understand what why the it's getting a higher Ah reading than it's expecting. The system can not reason that that if it detects more amp-hours than the it should have that it has that much more charge time. It just sees "ERROR" and since you're dealing with a system function that is designed to prevent loss of data the default action is to shut down to prevent loss of data.
You may be able to fix by resetting the Power Management Options. Unfortunately, with out doing test on the machine, it is impossible to tell. I could be software or hardware. If nothing else works, try replacing the battery again.

Do you still have the old battery. If so charge it, and see if it charges properly. If so then you have a bad battery.

it could be that your CPU ( central processing unit ) need new thermal compound. if you do not service your laptop every 3-6 months you will blow the cpu if the thermal compound gets to hard. the pc shuts down wen the cpu is to hot or if there is no conduction between the cpu and the heat sink

The processor is designed to gear down when you switch to battery mode to conserve power, this way the rechargeable battery last another 30% longer at least.

First make sure your fan is operating on your processor it takes about 3 mins for it to over heat and stop and shut itself down for safety to prevent burning up the processor.if its not this read on.
Also when you boot into safe mode then do a system restore to a date before the problem started. But first in safe mode run your antivirus scan to find something that is missed while being hid behind the operating system while functioning. Remember to rate my solution and leave testimonial this is how i earn my achievements here on FixYa thanks in advance

There are a number of reasons that may causing your HP computer running slow to shut down or start up, but generally two factors: outdated hardware or poor system maintenance. For hardware part: you can get more RAM, higher CPU etc. For system part: there might be a lot of junk files, caches, errors etc out there, about 80% slow HP issues are caused by this reason, in this case you can rely on http://www.speeduphp.com/ to free detect and fix related issues. Regards!

As you say the laptop handles the video well in the beginning, then when it runs for a few minutes it has trouble rendering the video in sync with the audio.
The reason for this problem is that 720p videos need the Graphics card in the laptop to work hard, this process generates heat and this heat needs to be removed from the laptop or the whole system does what's known as throttle down.

It slows down to reduce the heat it produces, thus preventing damage to the system.

The solution to this problem is to clean as much dust out of the laptop computer as possible, use a vacuum cleaner. Turn the laptop off and remove the battery. Clean the vents, it is a good idea to raise the back underside of the laptop with a pencil or the like. Some-thing that will raise the laptop off the work surface without obstructing the vents on the underside of the laptop. By raising the laptop it will allow for greater airflow through the laptop, therefore reducing the operating temperature of the laptop therefore making it more efficient and responsive.

Your laptop shuts itself down, this confirms that it is overheating. It does this so it does not damage itself.
If it still overheats then you will need to contact a computer technician to give it a good through cleaning. To clean the dust between the heatsink fins. Possibly increase the fan speed or replace the fan.

Take care!
damnor01

check ur hd for bad sectors...shutting down for no reason is usually this(bad sectors)
also check system information and check device manager..could be software problem/incompatabilty/no drivers installed/or two devices trying to use same resource and shutting down cos cant.

FIRST REPLACE YOUR BATTERY, THEN ALSO CHECK VIRUS WITH AN UPDATED VIRUS CLEANER.

does it just shut down randomly? and then when you turn it back on it shuts down nearly straight away? if so then it sounds like the processor is overheating and as a saftey mechanism it shuts down. which could be a big problem.

or you just need a new battery. because you are supposed to only charge it when it is dead. if you keep it on charge and plug it into the mains randomly,then the battery will eventually lose it capacitence (basically how much charge it can hold) and then it wont last long atall. you want to hope its the battery.

hope this helps.