I've been using the free version of Avast for years with no problems. They have a free version you can use permanently, you just have to re-register each year. If you were using a free trial of a paid version, just go to the actual free version. I don't have any personal experience with Kaspersky, sorry.
As Entomber said, paid versions are preferable to free but what's more important is user education. Users are usually hit by viruses because of malicious emails, visiting malicious sites, downloading infected files, or using infected media. You don't need expensive anti-virus software to avoid these things. If you can't identify where an email came from, don't download attachments or click links in it. Malicious sites are harder to avoid because you really wont know but the vast majority are porn, so avoid that. Avast free has a web-shield that will alert and stop you if you try to visit a malicious site. If you have to download files, manually scan each and every one of them before you open them. As for media, don't stick USB drives into your computer unless you know where it came from. On top of that, set up your antivirus to do system wide scans monthly (at least).
I've never used Kaspersky but I have had to clean PCs of user that did (using their paid professional versions at that). No matter how good an anti-virus may claim to be, they can be circumvented. Its better to just maintain good habits.
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