Is there a word to describe someone who does nice things for others, only to make themselves look or feel good?

I would rather use hypocrite for such a person as they are actually different privately than they struggle to convince people they are.

Hypocrite:
2) a person who feigns some desirable or publicly approved attitude, especially one whose private life, opinions, or statements belie his or her public statements. - Source:[Dictionary.com]

This might be difficult to pin down, but this might be close: Self-satisfying

Reference.com has it as:

Self-satisfying

adjective

1. effecting satisfaction to oneself.

Consider, image-conscious

Concerned about the general impression that one presents to other people. ODO

I immediately thought of pretentious.

1. Claiming that or behaving as if one is important or deserving of merit when such is not the case: a pretentious socialite.
2. Showing or betraying an attitude of superiority: made pretentious remarks about his education.
3. Marked by an extravagant or presumptuous outward show; ostentatious: a pretentious house.

[The Free Dictionary]

Consider selfishly beneficent

Beneficent
(of a person) generous or doing good.

and

Selfish
(of a person, action, or motive) lacking consideration for other people;
concerned chiefly with one's own personal profit or pleasure.

Depending on the level of selfishness or consciousness towards their behavior, you might swap selfish with non-altruistic:

Altruistic
Showing a disinterested and selfless concern for the well-being of others; unselfish.

Opportunist - Opportunists are people who see a chance to gain some advantage from a situation, often at the expense of ethics or morals. An opportunist seizes every opportunity to improve things for himself.

Self-aggrandising, perhaps (to increase one's power, status, or wealth (OED))

I'd lean toward one of the senses of false:

5. not genuine; counterfeit.

or fake:

13. designed to deceive or cheat; not real; counterfeit.

I'd use it to modify another adjective if I were given two words:

• falsely accommodating
• falsely chivalrous
• fakely polite

Either can also be used on its own as well:

• She is false (but may be archaic, see below)
• She is so false (seems more modern somehow)
• He is fake / He is so fake
• He is a fake

...but it doesn't say what about the person is false/fake. Using false in that sense may also be somewhat archaic; the phrase "false person" seems to have peaked in about 1823 and to my ear "She/he is false" sounds quite Austin-like. :-)

• disingenuous

oxforddictionaries - disingenuous : Not candid or sincere, typically by pretending that one knows less about something than one really does

ie. they are being disingenuous by feigning their concern for others

It might be a bit broad in its precision here but is not incorrect.

"Patronage"

"Patron"

It is a word(s) with a somewhat different meaning these days but historically a patron was someone wealthy who supported the arts or artists. It is something that has generally come to mean altruistic in todays usage but altruism is a separate thing from patronage which would have been seen as a symbol of power, wealth and even apparent kindness.

Where an altruist would provide food to the poor, a patron would commission a famous artist to paint a pretty picture for their feasting hall. While it doesnt specifically refer to the callousness of the "altruistic" act, patronage is used to refer to kindness that is very visible and that returns some sort of boon for its patron.

In just a single word it is difficult to find a precise match, I single out "patronage" simply because it encompasses all of the ideals of being outwardly generous while really it returned plenty of quid pro quo and was often used in the fashion which you describe.

It does not however cover the description of the underlying motivation of doing something good to only look good.

"Pretentious"

This comes close but that is more about embellishing the truth.

"Pander"

"Pander" is also close but it means to do these things to please others rather than solely to effect peoples opinions of the person, though that is a grey area.

Assuming your audience is familiar with the Judeo-Christian ethic, the word "Pharisee" would convey the idea you're driving at.

Jesus portrays the Pharisees as impatient with outward, ritual observance of minutiae which made them look acceptable and virtuous outwardly but left the inner person unreformed.

[Wikipedia]

You can use the word genteel in this context. It primarily describes gentlemanly/lady-like behavior but with the connotation of being a facade.

: pretending or trying to have the qualities and manners of people who have high social status
_{Merriam-Webster}

I often associate genteel with southern hospitality, which is sometimes criticized as being largely superficial (although I personally give individuals the benefit of the doubt until proven otherwise).

Southern hospitality has been examined by sociologists and other social scientists, sometimes characterizing the practices as designed to cover deficiencies in southern culture, such as slavery, discrimination, and widespread poverty.
_Wikipedia