Android - How to remove a Virus that installs an unlimited amount of APPs causing the smartphone to stop once an internet connection has been established?
As I already mentioned in the comments, here are the primary points you need to be careful about:
Whenever you install an app from the Play Store, you should pay attention to things like permissions, ratings, reviews by existing users, etc. before installing it on your device. One malicious install, and your phone could become compromised. It doesn't matter then, whether you are rooted or not.
When you install an app, it asks you a bunch of questions like what all things it needs access to (like SDCARD, telephony resources, system tools, etc.). Even without root access, a malicious app can do lots of damage to your phone, if you permit them these resources.
As the existing answer suggests, rooting is your only option if the malware has landed itself in the system area. But if you go along that route, proceed with care and caution, as there is a risk of bricking if incorrectly done.
For future reference, make a mental note to double check at least two things before installing an app: 1) The permissions it needs are sane and safe. 2) There are at least 500 million installs already with at least a 3.5+ rating. You can make rare exceptions for popular apps like WhatsApp/Skype, but this is generally what you should do. You may or may not be able to recover this phone, but life goes on and these things will go a long way to ensure that such a situation will never arise.
Depending on your phone brand, flash stock firmware through Odin, fastboot, or LG Flash Tool to overwrite the virus.
Samsung phones: Hold the power button, home, and volume down to access download mode. Download Odin v3 or any other version, I recommend XDA as a source. Find the .tar file for your device, SamMobile.com has all the firmwares for every Samsung device.
LG phones: download LG Flash Tool online, and find a .kdz stock firmare for your device.
Nexus phones: find the fastboot.exe included with your adb drivers and use that.
You didn't specify model or device so my help is only so limited here.
Anyway, hope this helps
EDIT:
Researched the Kazam Tornado 350, and there are no development threads for it on XDA or other sites. Couldn't find any stock firmware for it on Google, only shows a couple hits for the Tornado 348. I hate to say this, but you'd be better off getting a new phone.
Try a reset, Boot your Phone into safe-mode (Click on the tag to find out how) this will only boot essential apps, Then uninstall any apps you think are the culprit, then boot out of safe mode and see if the problem persists. If it does Repeat the cycle till you find the problem. Then by the end if the problem still persists it may be a system app (Due to me not finding much info about your phone, I can't say it is full of crap ware, and finding info about this phone is difficult)
But due to your broken screen, this may be difficult.
And due to the extensive conversation in the comments section above, I doubt that your phone has root privileges, but if the phones does have root privileges, and the application may be in the system folder.
But due to my limited knowledge of what you have on phone, what you have done to it etc. I can't be any more help, then just suggesting to boot into safe mode.
In my opinion I think it is just one of the default apps downloading use crap or the default apps have automatic updates enabled