Am I making a mistake trying again with my partner after everything hes done? Can I forgive him? Fresh start??

My boyfriend is a liar, proven. Hes also an alcoholic. He asked me to marry him then walked out 6 months later. He left me with 2 children and took his 3 (my step children who I love like my own), an eviction notice and not a cent to my name. I had to move in with my sick, pensioner Dad, lost my job and my children had a hell of a time moving schools etc. Ive been seeing him again for about 2 months now but Im not sure whether Im making a huge mistake!!!!! I love him to death and havent been able to breath without him, but sometimes, Im sure I deserve better than what hes given me in the past! Hes been trying really hard to stop the lying but I still cant trust him! We fight bout stupid stuff coz he just wont listen to how I feel and it always ends up in a big argument. My children adore him and love having him around, hes been amazing with them and loves them like his own too but is that enough to keep us together????