My iPhone is randomly making text messages disappear

Can I access my imessages through my Dell computer? Somehow, my iphone is randomly making text messages disppear and i need to access them. I'm signed in to my Apple ID/iTunes on both devices. i know the sender isnt deleting them through a third party app, because they've tried sending the same message to me several times. I don't know what's wrong; is it my service?

[Re-Titled by Moderator]