1) Programs won't start or crash suddenly: If some system programs like the task manager or registry editor have suddenly stopped working or crash randomly without any visible reason, it is more than likely that your computer has got some form of worm that is attaching itself to various programs and thus, corrupting their code. A virus modifying registry entries to prevent these programs from running could also cause this.
2) Unusual Network Traffic: If even after closing your browser and other applications that may use the internet, you see that your computer is still downloading or uploading anything, then your computer is possibly infected with some form of spyware that is either downloading other viruses to your computer or is uploading your private information to a remote computer. In case you use a wifi connection, it is also possible that your neighbors are using your network. If that indeed is the case, you need to make sure that wireless network is properly secured.
3) Internet Connection not available or is slow: If suddenly, your internet connection has stopped working or has become extermely slow, it is possible that a malware is the cause. However, problems with your internet service provider could also be the cause of such issues.
4) Annoying Popups or Redirection to Strange Websites: If there has been an increase in the number of annoying popups while browsing the internet or your web browser redirects you to a different website from the one you wanted to visit, a virus or adware might be the cause.
5) Computer won't boot: If your computer has suddenly stopped booting, it is possible that some malware has modified your computer's system files. In such a situation, re-installing or repairing your operating system might be the only solution.
6) Slow Computer: Some viruses are designed specifically for this purpose. Malware programmers achieve this by designing viruses that increase your CPU's usage level to 100%. If on running the Task Manager, you find that your CPU's usage is constantly at 100% level, a malware might be the cause.
7) Programs start automatically: Some viruses and adware automatically start strange programs on your computer in order to show unwanted advertisements or simply to frustrate users.
8) Unusual File Associations: Some viruses change .exe or other file associations which prevents these files from opening. Some viruses also do this to automatically start themselves when a file of a particular type is started.
9) Security Software Disabled: Most malware disable antiviruses and firewalls to prevent themselves from being detected. If only a single software stops working, it might be a specific software failure issue. If it is a general trend and almost every security software is facing this issue, a virus infection is most likely.
10) Weird Behaviour: If your computer automatically starts sending emails or opens strange applications randomly or does things different from its "normal" behaviour, your computer is most likely compromised by malware.
Most malware infections can however be avoided by using a powerful antivirus software and following some best security practices.