If your rough patch was so bad that you MUST address it in your application, your best move is to be honest (but brief!) about what happened, demonstrate how you've moved past your mistakes, and focus on your recent achievements.
I speak from personal experience. My second year of undergrad was rough. I was suicidal, diagnosed with a mental illness, abused my medication, failed classes, cheated, the works. I was caught cheating and received essentially the maximum punishment that wasn't suspension: I had to fail the class, and there would be a permanent note on my official transcript that says "This person received a mandatory fail for class XXX due to violation of the academic code." It was a huge wake-up call. I worked my ass off for the next few years, pulled my grades up, and ended up getting into my top choice Master's program (Ivy!)! If you told me that in undergrad I would've laughed because I thought no grad school would accept me. Here's what to do:
1. Mention it only if you have to.
Only mention your mistake in your application if it's visible to the admissions committee. In my case I had to because it was in my transcript and NOT explaining the note would be a huge red flag. If whatever you went through isn't obvious from your other application materials, then mentioning it is unnecessary. The admissions committee only skims your application, and something you think is horrible could be something they don't even notice. Can you give me more details about your situation? What exactly happened that was so bad that you're considering explaining it in your application?
2. If you have to mention it, be honest but brief.
Don't dwell on your mistakes; a few sentences of explanation are sufficient. You want the committee to focus on your achievements and not your failures, so just mention the negatives as straightforwardly, concisely, and professionally as possible and move on. Don't give more information than you have to and don't get too personal. In your case I would not mention the mental illness unless it's absolutely relevant. I didn't even mention my bad grades; I just explained the cheating. I didn't want the admissions committee to remember TWO negative things about my application.
3. If you violated a moral or legal code, show remorse and show that you've learned from your mistakes. Demonstrate how you've moved past your old mistakes onto future successes.
The point of disciplinary action is to teach you a lesson. Show that you have learned from your errors. End on a positive note by mentioning how you've overcome your mistake to achieve your recent successes. Show that you've moved past your mistake and ready to work hard in your dream school's grad program.
4. Submit the explanations document separately from your personal statement if possible.
Your personal statement should be overwhelmingly positive and confident in tone, and anything negative in it will be jarring to the reader. If there is a separate section in the application for you to submit this explanation document, then do that. If they don't offer a separate section, contact the school and ask.
5. If there's someone with clout who can write you a recommendation letter, ask them to explain it for you.
This person should be pretty influential, though -- it should be someone the admissions committee can trust. In my case I got a professor who was on the admissions committee of the school I was applying to to write me a recommendation letter. I had taken a class with him as a visiting student, and when I asked him for advice about explaining my academic violation in my application, he offered to explain it in his letter. This is something I would do only if your letter writer is someone the admissions committee trusts more than they trust you.
Admissions committees are people and they're willing to forgive you. They were students once too.
Congratulations on your recent successes! Overcoming mental illness to graduate with honors is an amazing achievement that you should be very proud of.
I hope I've helped. Feel free to contact me privately to discuss your situation in more detail, if you'd like. I spent a lot of time and met with a lot of professors to figure out how to address this in my application. I'd be happy to share what I've learned.
I should mention that I'm in the US and studying computer science. My academic violation was in an art history class unrelated to my major.