You want to use the word "check" with its regular meaning, so "I'll check the computer" or "I'll check the billing database" is fine. It's important to understand that «phrasal verbs» often have a very different meaning from the regular meaning of the verb and preposition. For example: "CHECK OUT these awesome sunglasses!" "You may not CHECK OUT reference books from the library." "I'll meet you for dinner after I CHECK IN at the hotel." "I would like to schedule a CHECK UP with Dr. Smith." These are actual phrasal verbs, or in the last case a phrasal noun ("a check up with"). You can find definitions to them easily on the web: http://idioms.thefreedictionary.com/check+in

It would be usual to say I'll check ON the computer...