When I introduce myself (in like a WhatsApp Group, etc.), does "Hi there, Chris here" works?
WhatsApp doesn't list the name of the sender of a message unless you have his or her contact saved on your phone. Otherwise it only shows the phone number, so introducing yourself is likely a big help. "Hi there, Chris here" is a perfectly fine introduction and is grammatically correct. You could also say "Hello, this is Chris" or "Hey, Chris here".
Also, don't worry about little intricacies like that. I also sweat over every single message I write, but it's unnecessary. Don't sweat it.
Good points in the comments. I would add that in this digital context, if one's handle/screen name/etc is not the same name that people know one by, it could be useful to include one's IRL (in real life) name. Also it could be useful to specify one's name to resolve ambiguity if you happen to have the same last or first name as someone else, combined with the aforementioned factor.