How are servants of the Many Faced God employed?

Although the quote is from book canon I believe it is the same in show canon. Essentially the Faceless Men are hired the same as any assassin is hired... with money.

Ned bowed, and turned on his heel without another word. He could feel Robert's eyes on his back. As he strode from the council chambers, the discussion resumed with scarcely a pause. "On Braavos there is a society called the Faceless Men," Grand Maester Pycelle offered.

"Do you have any idea how costly they are?" Littlefinger complained. "You could hire an army of common sellswords for half the price, and that's for a merchant. I don't dare think what they might ask for a princess."

[...]

Littlefinger shrugged. "Titles are cheap. The Faceless Men are expensive. If truth be told, I did the Targaryen girl more good than you with all your talk of honor. Let some sellsword drunk on visions of lordship try to kill her. Likely he'll make a botch of it, and afterward the Dothraki will be on their guard. If we'd sent a Faceless Man after her, she'd be as good as buried."

_{A Game of Thrones, Eddard VIII}

In the case of the TV series, I'm not really sure if another actor really paid The Faceless Men for this task or not, because Lady Crane is perfect obstacle in Arya's advancement...

Lady Crane is playing Cersei Lannister and is probably the best actress in that troupe of mummers. Viewers also know that Cersei is one of the names on Arya's list and that one of the Faceless Men that used Jaqen H'ghar's face before seemed to read Arya's mind by naming off names, meaning he knows...

JAQEN: A girl has many names on her lips: Joeffery, Cersei, Tywin Lannister, Ilyn Payne, The Hound, names to offer the Red God. She could offer them all, one by one.

Arya forms a quick bond with Lady Crane, who really has dark colored hair and almost behaves like Arya's mother Catelyn Stark, who was murdered by Walder Frey. In the scenes where Lady Crane is caring for Arya after a fight with the Waif, the mothering aspect becomes more apparent.

One has to ask how anyone in that theater company could afford to pay for this assassination and/or if the Faceless Men do any of this assassin work to their own political benefit??

We understand that it's part of their Religion, taking life to offer The Red God, but we do not know much else including for what purpose/to what end they exactly do this and if there are instances of assassinations any given Faceless Man (alone or together) will not do?

The Faceless Men convene in a chamber in the House of Black and White to discuss potential assignments and dole out contract assassinations. They discuss the potential deaths in the Braavosi language, though some may speak in High Valyrian. Debates can become very heated.[4] An assassin only accepts an assignment if they are unfamiliar with the target

How are these decisions made, because clearly there is a process of induction, which means some decisions by "someone" have to be made. (Arya is turned away, and then later she is accepted, and from there it is debated if she can make it) Another example is the choice to let Arya become Arya again and to leave with either Faceless Man knowledge and/or metaphysical/magical abilities.

In addition this plot or idea of an actress wanting to kill another is a deep irony considering Faceless Men basically impersonate "someone else" like an actor/actress does, calls back to the jealously/fighting between Arya and Sansa, is a juxtaposition to the Jeyne Poole book plot, and Arya's advice to Lady Crane on how to play Cersei Lannister reveals that Arya actually understands Cersei (again playing on a theme of motherhood, as it's Cersei's only redeeming quality and the scene Lady Crane is playing deals with a reenactment of Joffrey's death), which is a profound reveal in terms of her character's ability to emphasize, despite that she still wants to kill her.

But moreover it comes off more of "a test" for Arya, because it turns out to be rather personal for Arya, unlike her previous task...

Other things to consider:

Although it is not clear how special Arya's case is, because we don't have any other instances of outsiders or insiders we can "trust", we do see from her taking the life of three other people to rescue "Jaqen H'ghar", that he owes her three lives. In this instance "money" is not a valued currency and suggest that there are other means for which any given Faceless Man can take life from others.

Like I was saying in the comments, there really isn't a lot of information on "how" the Faceless Men really work as an organization, outside of Arya's brief inside view and the books distinguishing through a few of them by looks and "roles" (The Waif, The Kindly Man, The Fat Fellow).

According to Matthar, the Iron Bank was founded by sixteen men and seven women who hid valuables in an abandoned iron mine shortly after the foundation of Braavos. As the iron mine's chambers filled with treasures, a bank was formed to utilize the wealth. Each of the twenty-three founders had a key to these great subterranean vaults, and their descendants—now numbering at least one thousand—are known as "keyholders" who proudly display ceremonial keys on formal occasions. Other powerful Braavosi own shares in the bank, sit on its secret councils, and have a voice in selecting the men who lead it.

Though all the Free Cities have their own banks, the Iron Bank is richer and more powerful than the others combined and has a fearsome reputation when collecting debts. When princes or kings default on their debts or are foolish enough not to honor their agreements with the Iron Bank, the Iron Bank supports new princes and kings to appear. These new princes and kings then honor the previous debt along with paying back the money the bank loaned them in claiming their new power, lest they suffer the same fate as their predecessors.

In addition there is some information to suggest that Braavos is also possibly not as it seems, as although there is Sea Lord residing over it's political affairs, the books tell us that the Iron Bank also holds a great deal of sway and that there are "keyholders" and "secret committees" whom have all the power. It's possible that The Faceless Men, The Iron Bank's Key Holders, and the Sea Lord/his family could all work in tangent knowingly or unknowingly...

With the ability of the Faceless Men to take on any face that come to have,one could argue that they too could have heavy political sway in either Essos or Westoros, depending on whom they could kill and impersonate. (Case and point: Arya killing Walder Frey and now having his face)

In the books, the waif explains how she ended up in service of the Faceless Men (A Feast for Crows, from the chapter "Cat Of The Canals"):

She did not expect the waif to answer, but she did. "I was born the only child of an ancient House, my noble father's heir," the waif replied. "My mother died when I was little, I have no memory of her. When I was six my father wed again. His new wife treated me kindly until she gave birth to a daughter of her own. Then it was her wish that I should die, so her own blood might inherit my father's wealth. She should have sought the favor of the Many-Faced God, but she could not bear the sacrifice he would ask of her. Instead, she thought to poison me herself. It left me as you see me now, but I did not die. When the healers in the House of the Red Hands told my father what she had done, he came here and made sacrifice, offering up all his wealth and me. Him of Many Faces heard his prayer. I was brought to the temple to serve, and my father's wife received the gift."

A Wiki of Ice and Fire has extrapolated from this and other mentions and come up with this description:

For a price, the guild will agree to kill anyone in the known world, considering this contract to be a sacrament of their god. The price is always high or dear, but within the means of the person if they are willing to make the sacrifice. The cost of their services also depends on the prominence and security of the target.