Anon, I'm sorry, but you have several problems here.
I turned on the computer to check, look up some things. Then chatting incidentally to an French.
I turned on the computer to check [to check what? you need an object. to check the weather, etc.], looked some things up, and then chatted with a person from France.
In other words : I turned on the computer to check, look up some things. Then I chatted incidentally to an French.
I'd prefer to use the gerund like the first sentence, There is no gerund in the first sentence
is it correct ? Or else, how to fix it with gerund ? Gerunds are not appropriate here. I like chatting on the computer is a gerund.
Thanks for your help. I hope it helps.