Anyway to increase these people to attack you? I've not only robbed several people, but also robbed a few venders that had the guards chase me.

When will I expect these group of people to start attacking me?

It's random AFAIK. I have seen people send them after me for things I don't remember doing. Hell the Silver Blood family sent them after me because I entered the mine in Markarth.

I remember having a group of hired thugs attack me because I pickpocket from a stormcloak guard. Now I want a group of them to attack me again. I want to see how great my new dragon armor can sustain a bunch of brutal characters

Steal from Drunken Huntsman in Whiterun. Keeper of Drunken Huntsman was only living NPC which I stole from with my Altmer character and I got thugs after me. I just took Imperial Bow from his table and he started to hire people to give me a lesson.