My characters are playing as members of a thief's guild. I have an idea for larger goals for a guild, but it is the sort of thing I would like to expand on more to build some intrigue.The guild has 5 branches with head members and then descending army like ranks down to the majority of their membership, their "fingers." The guild has wealthy patrons and high ranking political connections within their city-state to the point of having underhanded control there, but little influence outside of the city-state.

Any ideas?

I only have the one suggestion:

Depends on leadership.  If they essentially have control of the state, they start using legal maneuvers to advance their business interests and harm their rivals.  They can try to expand their business to other cities.  I mean basically, what's making them money right now?  Because the grand scheme is going to be to make more money.  If they only have the one city-state, they'll start looking for a foothold in the next one.

There are any number of real world historical and current examples of governments captured by Thieves' Guilds on both sides of the Atlantic.  They start extracting value out of the nation and funneling money towards themselves.

In historical context, any thieving guild/gang would have the main objectives of profit, secrecy, and reputation. They want as much as they can get, they want to remain as ahead of any hunting government/organization as they can, and they want a reputation that makes them reliable to patrons, good for business with other guilds, will attract new members, but will also tell others they aren't to be messed with. 

With the organization you've described, there is a very clear opening for plot; having 5 head members instead of 1 overall leader. While having 1 leader means that all the calls/orders will be to their standard, having 5 means that they may not always be on the same page. No matter how close, a group of people will always have differing ideas. Maybe one leader thinks the guild should add a new business venture (maybe certain goods or smuggling) that the others do not agree with. Maybe 2 or 3 have blackmail on someone important in the world and the other 2 want to use it. Maybe they have different moral compasses, and often argue over the next course of action.

All in all, you have a lot of opportunity to create strife at the very top of your guild's power structure. That kind of thing trickles down the ranks and will effect every aspect of the guild's ability to fulfill their main objectives. Use it as you will, hope this helps.

After enough time has passed since its creation, one of the main goals of any organization becomes self-preservation (cynical, I know).

I’d think this would be doubly true with a group that exists with the stated goal of robbing people. I’d say most of the guild’s actions would have the end goal of preserving the guild. They probably know just how far they can push things before the powers that be decide to push back. They know who they can steal from and who they must leave alone. They know how often they need to let one of their members get caught (likely someone who’s fallen out of favor) so the authorities can give the appearance of trying to stop them. Whatever they are doing, you can bet the leadership’s priority is going to be not to upset the status quo. 
But within that, there’s got to be a fly in the ointment or there’s no story. So, like atlas is saying, do a power play. One person (Or a couple) decides they want to be in charge now and starts doing things to intentionally disrupt the situation. This can be even more interesting if you have three factions to play off each other. Personally I find four or more to be too much for people to keep track of, but maybe it could work at your table. Let the players decide which side they want to align themselves with, and then have at it. 
I could see a cool NPC role in this for a city watch investigator. They could become acquainted with the PCs and try and force them to cut deals like turn in some other thieves or they’ll arrest the party. They could take an opponent off the board, but need to make a deal with the devil to do it. That would cross a line for the guild. It’s one thing to fight between themselves, but no one would be willing to work with the watch, and could turn them into enemies of all guild factions — if they get caught.

Depends. Usually to gain a lot of influence of a major city. Then create little branch offices to other large cities so that they have a lot of friends in very high places. Sorta giving them a protection bundle for coin. Securing very valuable items which helps them gain more power. Pretty much that. 

Other goals could be if favorable relations to the party, slipping them out of jail when they decide not to murder their way out of a guard arresting them scenario. Or when an encounter is too tough and the enemies totally decided to knock them out, they can be slipped a dagger and a lockpick. If unfavored, maybe disrupt a long rest when it makes sense for assassins to be there. Especially if a bit of a lead up and they would have reasonably survived behind them. Give hints or a chance to detect them ahead of time.

Look to the Godfather movies.  There are two great scenes where several 'families' gather to discuss 'business.'  Vito Corleone lays down the law for his family not to be involved in the least way with any drug trafficking.  And it is shown that organized crime didn't have one supreme figurehead, but they were an unofficial confederacy.

Initially the goal would be to find a method of making "profit" and staying out of jail; a way to execute the 'perfect crime.'

The next goal would be to establish your turf with the other nefarious groups.  This would be accomplished by doing favors and 'defending' your turf.

The third stage would be to expand your turf and diversify your 'rackets.'  This would be accomplished by doing more favors and improving your organization, particularly the 'unity of command.'  It is at this stage that the Guild starts to look like real organized crime.  The guild will begin influencing law enforcement at this stage, corrupting 'police', DA's and judges.  They have also established their own internal Code of Conduct that is enforced ruthlessly against members that are careless or just stupid.  For example, No kids are harmed or allowed to see what we do.

The fourth stage will be to essentially establish themselves as their own government within the country.  At this point they are so effective that they have immunized themselves from the countries laws.  They are most certainly already expanding their influence into other countries.

The key motivation of a thieves guild and its members is profit. They are in it for the money. With money comes power and influence but those are only useful in the pursuit of more money.

Guild leadership may get carried away with the power that the guild acquires as a result of their money making activities. Some will want to take control of society. Some may want a air of legitimacy and the ability to move with the upper levels of society on an equal basis. The leaders may begin to act as lords or other nobles, considering the other thieves to be simple henchmen expected to follow orders.

However, a thieves guild is not typically a place of honor or fair competition. Every member of the guild is in it for themselves and for the money. Some may need money to support families, others want it for the finer things in life or have a specific goal in mind that requires funding. By nature, a thieves guild is pretty lawless, they don't necessarily follow rules even the ones laid down by the guild. As such, guild leadership is always a bit insecure, others may seek to move up within the organization either by setting up someone higher up to make a mistake so that their peers remove them or by direct intervention to remove them. Guild leaders may pretend to be lords but they often have assassins and competitors lurking around corners waiting for them to make a mistake.

In any case, the goal of the organization is more profit, the leaders are also motivated towards maintaining their power and the power of the guild to support that. The guild itself is just another power structure that enhances the ability of the guild to make money for those at the top and its members. If the guild isn't profitable for its members by helping earn income and providing fringe benefits like protection from law enforcement (for the more powerful guilds) then there isn't much point for the individual thieves to be members. Trying to keep individual members in line through threats and force can work in the short term but in the long term the leadership ends up replaced if the guild isn't also of use to its members.

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In your case, the guild sounds like it is well established in this one city state but it also seems that it has reached a saturation point in the local market. There are only so many things to steal, so many victims to be harnessed, if the thieves guild is too active then (unless the entire city is actually the guild and comprises a significant fraction of the population) various factions will organize very powerful active resistance. So it is in the best interest of the guild to maximize profit without getting too large (or get so large that they completely replace the government - but this will likely lead to unrest and revolution down the line since a thieves guild government may not be up to providing for the masses to keep them under control).

In such a situation, the guild will look for good well paying opportunities both in the city and outside. The will keep politicians in their pockets and try not to be such an inconvenience that the local populace and uncontrolled politicians don't organize an uprising against the organization. The guild would also seek to expand their influence, setting up satellite guilds in neighbouring states. This will increase profits by having more markets to raid but in addition, as they acquire political power, they will be able to avoid wars and other national interactions that would be damaging to their business.

Motivations: Profit, Expansion, Stability and Security.

One of the difficulties with the D&D 'thieves guild' is that they don't precisely correspond to any real world organization. It appears that D&D thieves are inspired by Elizabethan-era criminal organizations (e.g. that's where "thieves' cant" comes from), and while organized gangs of thieves most certainly did exist, it's unlikely that they had the persistence, organization, and vague social respectability attributed to D&D thieves' guilds. The Mafia is probably a better model for D&D thieves' guilds.

While some underworld organizations (even ones that control extensive criminal networks) do have political or social objectives, such groups are probably going to be called (depending on who's speaking) freedom fighters, revolutionaries, or terrorists; if it's called a "thieves' guild", it probably doesn't have major objectives other than money and power. This doesn't produce all that much of a 'long game'.