I am in the US and I have a BBA (Bach in Business Admin). I am going back to school to get some psychology courses under my belt before I apply for Clinical Psych Ph.D. programs in the US. I met with a Ph.D. student and she mentioned that there were a few people in the Clinical Psych program who had degrees in other disciplines. I think she mentioned that one of her colleagues had a BA in English or something totally unrelated to Psych and still got in. So it's possible! I think the key is getting relevant research experience, being clear on what your research interests are, etc.. But that is all for a Ph.D. If you are looking to do a master's, it might be easier to get in given that you already have psychology courses and all that.
Also, you do not need a master's to get into a Ph.D in the US.