It's interesting, and very kind, that people constantly inquire after your welfare once you have lost a partner. The professionals whom you have to contact about all the unpaid bills and the other legal miseries that inevitably come bobbing along in the wake of death pretty much universally use the phrase, "I'm sorry for your loss." This comes either early or late in process of taking care of business. My response is usually, "Thank you, so am I." It's sincere, but somewhat distancing. A statement of fact, it only requires a quick response.
People who know you more informally, however, take it farther. They ask, "How you doing?". This is a response that triggers thought, because it more or less implies that they're aware that you're going through a difficult adjustment period, and it's definitely more supportive in its intent. It's more nuanced than "How are you?", and less insouciant than, say, "How's everything going?" Its implied focus is on you and your particular feelings and situation.
But "How (are) you doing?" is difficult to answer, because sometimes you just want to stop for a minute and say, "Well, I'm glad you asked. I'm having a lot of trouble figuring out exactly what's going on with our bank accounts." Or maybe it's more like, "My God, I didn't know chicken got so dry and tough if you left it in the oven too long." But does anyone really want to have you do data mining like that? In the computer world, we call that a core dump. Probably it's best to keep it brief, honest, and appreciative in tone.
"How (are) you doing?" can be pretty open-ended like that. It's nice. It presents one of those rare opportunities that if you feel like elaborating, you may, but you don't have to. Nobody has a particular agenda, other than to show an interest, and offer an opportunity.
So if you happen to see me, and happen to say "How ya doin'", don't be surprised if sometimes I say, "Just fine, thanks", but other times, "Well, the cat threw up on my kitchen chair, and a raccoon chewed a hole in the swing seat, and I don't think I like the way I cook, but generally, not so bad, thanks." After all, you asked for it. And thank you for asking.