Guy here. Why the heck are you dating someone who you say lies to you and you suspect of spying on you. That's unfathomable to me - it's a voluntary relationship, and if someone is so jealous and insecure that they really are spying on you and lying about it, it's only going to get worse with time.
I would change my passwords to something unknown (no pet names, children names, or birthdays) and enable 2-factor authentication on all of your account immediately. It won't help you catch someone spying, but it should make it much more difficult. I suppose you could add some false information, like talking about a trip to someplace that doesn't exist, and see if he brings it up.