While there are some good suggestions here, I think you would have the best chance of success not by applying as if you had no history with the company, nor by applying as normal, but mentioning the previous offer in a covering email.
You could consider approaching HR, but personally, I would approach someone who interviewed me, especially if I would be reporting to them.
I would want my soon to be new boss thinking “we interviewed this guy last year and really wanted him. Let’s not risk losing him again. Let’s get him on board ASAP”.
I don’t know your profession, but as a software engineer, I often see us hire good people, so as not to lose them to the market, and worry later about where to use them.
Since the company currently has openings that match you, I am sure that they will want to have you on board.
Just make it a friendly “hello, remember me? I interviewed you last year and you made an offer. Unfortunately, by that time I had accepted something else, and felt hon(u)r bound not to let them down. Now I am free and notice that you have some openings similar to the position which you offered me last year, and wondered if you would still be interested in me”. That should set things rolling nicely.