How do I fix my Lenovo A606 4g smartphone

we cant see your screen, so answers are very hard lacking photos of failing screens.
NO PC stated at all just Laptop ,it has model name and number.
NO OS stated, just windows, (XP?) the virus magnet XP?>
yes remove the huge battery first. run on AC PACK only , good now?
yes, clean PCs vents of all lent.
are you running said laptop on blanket blocking air vents, bad that is . do not do that.
funny screen graphics, and worst stalling is what to diagnose.
try this big battery out.
turn on PC, and go to the first BIOS screen using the BIOS hot key that is in your manual, I cant tell you, as no PC stated.
The if this screen looks good for hours on end., the Hardware is mostly ok.
if this fails.
does the fan blow air,? as it must for sure if it overheats the fan blows faster on all PCs like this.

all PCs overheat if not serviced (cleaned)
all do.
if fan dead
if fan can't speed up, due to lint packing IT.
vents clogged, and more.
first run with that battery pack larger removed, on AC only.
see if it run on AC , you can do that forever if you want,

if PC fan acts up say so or if dead.
my guess to is the GPU chip gets too hot.
The CPU self shuts down at 90C, by the magic of Intel Inside.
(or AMD inside )

If you are running Windows 7 or 8 there is a power setting that you can change to tell it when to shut down the screen or go to sleep. Click on the power icon in the lower right task bar. You will see something like power settings. Also you will see when to shut down the screen, for both battery an electric. You can make changes there an fix problem. I just did this a week ago an fixed the same problem.

Here are two things to try:
I- Using the Power button:
1 Press and hold the Power button for at least 4 seconds.
2 Swipe-down the shutdown confirmation screen to the bottom
edge of the display to turn off the tablet.
Using Charms:
1 Swipe-in from the right edge of the display to access Charms.
2 Touch Settings
Turning Off the Screen
Using the Power button:
Press the Power button to turn off the screen.
Using Charms:
1 Swipe-in from the right edge of the display to access Charms.
2 Touch Settings
NOTE: If your tablet is in a standby state (tablet is on, but screen is off), press the Power button or Windows button once to turn on the screen.
12
'
Setting Up Your Tablet
? Power
? Shut down to turn off the tablet.

II- Unplug the battery, wait 30 seconds, then install the battery again.

this might be a serious error on your mobo,chip failure etc

it may be due to network lock, contact service provider , else you can get unlock codes here http://www.onlinegsmunlock.com/
you need to specify the phone model and network provider

Your phone is set to work only with
certain SIM cards. Check if you are
using the correct operator SIM card.

Check that the SIM card is properly seated and locked down. Also clean the contacts of the SIM and the unit with 99% isoprophyl alcohol and carefully re-assemble.

sounds like the mother board maybe on its way out - is it under warranty?

This will fix it:

Turn the phone off, remove the battery for a minute. Replace the battery.

Hold down the right and left soft menu keys simultaneously then press the power button for a couple seconds and release. Keep holding down the soft keys until the hard reset screen displays.

The reset screen will tell you to press 'Send' to reset or any other key to cancel. Press and hold 'Send', which is the button with the green phone on it. After a couple seconds, the device will start to do its thing.

This will take some time so be patient. It will have a question or two as it goes through the process. Answer.

Don't worry, you won't lose your branding (e.g. if you are a tmo customer, you don't lose the ability to have myfavs and your phone keeps the tmo branding/settings).