A very good question to ponder..many thoughts come to my mind. Apologise for the rather long response!
Charity - is a selfless art of giving something you possess to a person in need. It is much better to provide the means for the people to gain long-term or long lasting benefit than a short-term final solution e.g. Give the man a fish and he eats for a day. Teach him to fish and he eats every day or something like that.
Hence, in my honest opinion, there may not be a question of whether charity in itself is selfless or selfish but we could consider charity as a means to generate more to be utilised for charitable purposes. For e.g. tax deductible donations. They can generate tax deductions which helps donor save a little more that can be (hopefully) reused for charity.
Honestly I have no idea of selfish charity - is it something you do through your own Trust or NGO to gain funds claimed as administrative fees etc. If that's possible that could be selfish - e.g. NGOs publish statements for donors to know how much of their $ actually reaches the beneficiary or is used for the intended purpose.
And when you intend to do charity - best to seek out legitimate, as-direct-as-possible funds, organisations that will not squander our little (precious for recipients) monies for purposes other than charity or where only 25% of the donation actually reaches the needy. There are many ways to do charity - almost always best to give in kind than in cash to avoid above.
Human mind always seeks simple outcomes than solutions - so sometimes money received could be misused by the beneficiaries themselves for short-term pleasures - hence giving in kind could help avoid that to a certain extent.
I do ALOT of charity work and volunteer work each week. Only the person themselves can tell you their specific motivation. For me personally, it is empathy and compassion towards all those elderly people I visit in different nursing homes on different days of the week. These people have become like family to me, and I tell them that. So, I get that love and affection that I give them, returned to me. It is very much a reciprocal relationship where we both get something positive out of it. Some people in nursing homes have no family, or family who live too far away to visit often. I have become very close to the families of the residents too. When they do come visit, they usually want to meet me and say "Thank You", which makes it all worth while.
I think it can go either way. Some do it to be seen or ensure some good "karma". There are other people who will do and not care one iota who sees what they do. I like the latter but each person must be convicted in his own mind on where he stands.
I would say that charity work or volunteering to help someone else is driven by both helping that person as well as feeling of service. Perhaps that is the beauty of the act. It not only benefits the receiving party but the giving party as well. It is of course known that people receiving the charity are benefiting from it. However, countless studies today are placing emphasis on the mental, psychological, and physical health benefits for the people presenting it.
I believe that it is initially driven by the idea of helping someone else, and then afterwards, finding out you are also helping yourself.