Hello and welcome to Beyond Blue. Please accept my condolences on the loss of your friends' father. The death of anyone in our family circle or our friends is always a shock and I understand how hard this period in both your lives.
It is difficult to know what to do or say, especially when you are also sad. For me the most important thing is to be there whenever your partner needs a shoulder or to talk about his dad. Often friends do not want to talk about the person who has passed on because they are afraid of further upsetting the bereaved person, and sometimes this true. Generally your partner will want to talk about his dad and about the past. Just listen and add a few comments if you are part of the reminiscence. Talking about the events of the past helps to heal the wound of loss and does bring comfort, so let your partner talk if he wishes.
Saying, "Let's not talk about this now." is depriving your partner of the opportunity to grieve and remember at the same time. It's good for both of you. I understand how hard this may be for you having to manage your own grief and feeling fragile because of your mental health difficulties. I have found that when you concentrate on another person such as your partner, your own problems take a back seat for a while.
Please remember you cannot take away your partner's pain. We all wish we could but this is not possible. So taking care of the day to day chores and keeping a comfortable place for your partner to be is one of the best things, however mundane it seems. Being sensitive to his moods and understanding that grief comes in waves will help both of you. The pain of loss comes then goes for a while and we think it's over. But then it returns and we go through it again. But each time it gets a little less intense and the waves get further apart.
There is no magic cure. Reassure your partner it's OK to to be upset and cry. Encourage him to talk but as I said, be sensitive to his needs.
I hope you will be closer together through this time until you both feel the grief has been acknowledged.