In October 2011 I visited my American boyfriend on the Visa Waiver Programme but for medical reasons etc. overstayed until May 2012. We want to spend more tome together and get married so I tried to get a tourist visa but was refused as I do not have significant ties with the UK.
Until October 2011 I had been living in Japan (for twenty years) so do not have a job in the UK. I am a British citizen.
How can I get to be with my boyfriend again?

He can move to Japan or UK and be with you ?

Are you planing on getting married, for a spousal visa ?