It depends a little on how you are detoxifying. Are you doing one of those 3 day or 7 day cleanses? Actually, that's NOT a good idea - way too much stress on your body, and then it doesn't "hold" because you go back to your regular routines that got you into that situation. If you are doing an ongoing detox program as part of improved nutrition, that's much better, but of course it depends on what you are using and how safe/maintainable it is.
The toxins in your body only have 3 ways of getting out - elimination (urine/feces), mucus membranes (especially nasal passages), and the skin. So break-outs are normal and not a bad thing. For the clogging, you can do a natural or a professional facial. The professional ones cost more, of course, but they are also softening the skin and opening the pores gently with steam. Some people do it themselves but you have to be very careful not to burn yourself. Anyway, removing the dirt and oil that's clogging the pores can be helpful.
The oil production from below the skin, and the toxin release from deeper in the body, aren't really helped by anything you do topically. That's all nutrition and internal balance. So it depends on what you are already doing and what you are interested in doing. I do the whole thing because I am strengthening my immune system, reducing inflammation (the major cause of disease) and working to prevent cancer. There's great science on this and I'm doing some cutting edge but very safe things. So my approach is whole body and while my skin is benefiting tremendously, I'm also never sick anymore and I'm allergy free. If I can help you meet your goals, let me know.
Good luck!