Your questioner provides a false duality, where there ought to be a trinity, since Christians will usually 'do good' as a response to God's offer of salvation, rather than as a means of winning favour with Him.
If this is the case (and I don't think it's as simple as that), then to do good only as a response to an offer of being saved from some terrible fate, seems less virtuous than doing good simply because you Love.
JJ's point above is important, I think. Does someone who does good do it for themselves, in order to feel good, or in order to make others feel good? I'd guess there's a bit of both in it, but that one is the main motivator. Doing good - acting in Love to help another, is something that can bring us the sort of deep fulfilment that we can't get from serving only ourselves, yet there is a paradox in that doing good
does serve us, in the sense that it brings us that fulfilment. I suppose, for an act of kindness to be truly kind, it must not only have a kind effect, but also be done from feelings of compassion, with at least some truly unselfish motive.
So what should we
do with our feelings of happiness gained from a good act? It's certainly a mistake to be all self-congratulatory even if we keep the self-congratulation to ourselves, yet ISTM that there is something very right in enjoying the outcome - the smile on another's face, if what we are doing is truly rejoicing in another's pleasure.
I guess those who have had experience of a relationship with children know best the feeling of pleasure gained from providing another person with pleasure. In general, we do not want to make happy times for children because we have calculated that we will get pleasure from their pleasure, in general - and I think this is true of most people, our real motivation is the feelings of deep compassion and Love that we have for them, and the pleasure we feel when we see a child laugh is a un-sought-after, but pleasant, by-product. But the fact that another's pleasure also brings us pleasure, is a wonderful thing. Such is the kingdom of Heaven on Earth.