Hi Sarah. In your example "onto" me is being used two ways. "I think they were onto me" refers to a state of awareness. It means that "they" understand what "me" is doing. If I was trying to hide something from you, but you knew it, you could say "I am onto you." Another example might be, "You are being nice because you didn't do the dishes. I am onto you!" or "The salesman said it was free, but I am onto to him." "She's been onto me to buy her a ring for half a year" In this example of "Onto" means to ride or nag. Such as she got onto the horse. So the speaker here is saying that "She" has been "riding" "nagging" "pressing" for "me" to buy her a ring for a while. Parents get "onto" their children to do homework, chores, eat right etc. Girlfriends get "onto" boyfriends to move in together, get engaged, get married, get jobs, etc. Hope this helps.

They were on to me = they know your up to something. She's been on to me about buying a ring for half a year = she has been bothering me about getting a ring.

be onto sb 充分了解某人..

If the second one WERE acceptable, it would be written separated as shown by Kaitlyn

Actually the second one is not right according to our (Australian) usage - in my opinion. More correct would be 'she's been on at me to buy ...' The second meaning is 'nagging' or repetitively