Having someone threaten to harm themselves if you leave (whether directly or indirectly) is emotional blackmail. They are forcing you to stay with them which is a form of abuse and imprisonment.
A good approach would be to make contact with a close relative or family member (or friend etc.) of theirs informing them of the fact that your partner is threatening to harm him/herself and requesting that they provide support to that person.
Speak to your partner and let him/her know that things aren't working out but that you will always be available to them on a "friendship" or support level.
If you fear that he / she may hurt you as well if you leave (there are numerous reports of murder-suicides worldwide), then do not break up with him / her in person or in person alone.