Chinese phrases to encourage better 关系
关系 has very little to do with language because it works more on a 礼尚往来-system, much like everything in China does.

If you want to "earn" some 关系 you must make people feel like they owe you.

The 礼 in 礼尚往来 originally stood for 礼节 - looking at it you might even think of 礼貌 or 礼仪 but most people today take this as 礼物.

If you want to earn 关系 then you really need to take the initiative and give some gifts, presents or even money. 

If you want to keep 关系 you better make sure you "reciprocate" on every possible occasion, especially: birthdays, births, funerals, weddings, promotions, house warnings. Reciprocation means that whatever you were given before you better give something with greater or (in the absolute least) equal value (but something of equal value will probably be looked upon as somewhat disrespectful and perhaps as an indicator that you're not willing to 'sustain' this 关系 relationship).

(On a side note: this explains China's problem with corruption - too many people owe people favors...)

If someone invites you to dinner - you better invite them back, if someone gives you a "gift" you better give one back (at the appropriate time, of course) - unless you don't care about "keeping up" the relationship.

If you recieve some stuff you can say something like:
破费了！

When you give stuff you can say something like:
一点心意