Your problem is social, not technical, so requires a social solution
From his point of view, you used a CD you brought in to make the computer run a different OS, and you had a network administrator's prompt on your laptop. If you appear to bypass his security, it's going to break his trust in you.
You can't solve this problem by trying to convince him there's no technical problem.
(Plenty of folk have been told by proprietary manufacturers that open-source software is a security risk, and they may well interpret the heartbleed bug as confirmation.)
Avoid winning the argument
What is a good way to convince him that I am not doing anything I shouldn't?
(Of course, first permanently stop doing anything he could possibly mistake for improper use of the IT facilities. This is step 0. If it's not necessary work that's done in the usual way, it's not what he feels the equipment is for.)
Unfortunately, it's very important to realise that proving you were harmless will involve him learning something from you during his disagreement with you, and he can't enjoy that, so proof will make the interpersonal situation worse not better.
Involving any authority figures (your parents, other staff) or any technical experts (web pages, any friends who also understand) will make him feel attacked or humiliated and you will provoke hatred - avoid!
Show you're a help not a hindrance
Change your aim to "What is a good way to convince him to trust me?"
You need to convince him by your actions and attitude that you're on his side and that you'll never be "a problem" again, then much later on once he trusts you, he'll be able to interpret your actions as having positive motivation, and will be able at some point in the future to believe that you never intended or caused harm.
It sounds like you used technical arguments to try to prove you didn't install malware, and as you see, he found places to perceive loopholes in your argument, because of your current role in his life: "smartass hacker".
You see innocent fun where he can only see problem behaviour.
How do you switch his perception from "smartass hacker" to "trusted helper"?
People have an opportunity to forgive when you apologise - apology is a powerful social tool.
First you need to apologise and show that you understand you've been a problem to him. You don't need to say you hacked anything to admit you've been a pain. Try something along the lines of
I've been doing a lot of thinking since you thought I'd deliberately put a virus on the computer, and I've realised I've been a pain showing off and messing around on the computers, and I wanted to say sorry and try and make things up. Is there some annoying, boring or long job that I could help out with so I can show you I want to help, not be a problem?
You might need to allow him, without interrupting and especially without disagreeing, to tell you what he thinks about what you did. This can be an important step in him expressing his emotions about it. Experienced couples learn to separate the feeling from the facts, using phrases like "When you ......., I felt .........". Don't expect this guy to be as careful, but giving him the chance to tell you off without you saying it was OK can really help him see you in a different light. Once he's done, you can say "I realise now how irritating the whole thing must have been - sorry again." Do not say at this stage that it didn't matter or didn't cause harm - it mattered to him, and you're apologising for the effect you've had on him.
It takes a while to win people round, and you'll need to be a good, reliable, endlessly supportive team member without any further unnecessary IT fun for a sustained period until he'll see you as reformed. Once he sees your motivation as good, he'll be able to doubt whether he was right to think you installed the virus and hacked the network.
What if he challenges you to admit to the virus?
Virus: "I can't see how anything I did on that computer could have installed a virus, but I admit I don't know as much as I like to say in front of my friends, and I totally see now why you don't trust me and so I'm not sure I can convince you I never meant any harm and didn't try to install a virus." - good because it positions you out of know-more-than-you land yet doesn't lie that you intended harm.
Can he ever want to believe it wasn't a problem?
[Much further ahead: Once you're at the point that the two of you can laugh about something together, you could share times you've played jokes on your friends, which may help him see that you were pranking them with the shell prompt, but lay off raising this event again explicitly. This is an already-thawed-relationship card to play without bringing up the unpleasant memories for him, but it won't work if you try to convince him - it just helps him put you in the prankster box instead of the hacker box. Crucially you can only go there once you're out of the problem box and into the fun-company box. If you never end up in the fun company box, don't try to move to the prankster box, because you'll just move back into the irritating-waste-of-time box instead!]
Most managers have to manage people in a wider sphere of expertise than the route that they took to being a manager, so have to deal with people who know more than them. Very few people find this easy to deal with, and people-smart employees know not to slap managers in the face with lack of knowledge.
Learning how to make people feel that you value and respect them when you who know more than them is a crucial workspace skill, and being on good working terms with everyone from the chief exec to the cleaner is tremendously helpful and has solved many many problems for me quickly and easily.
See it from the other person's point of view, and be nice. People will want to help you out because you'll be grateful.