How is the lottery of caskets in The Merchant of Venice a test for the suitors who have come to woo Portia?

Portia's father has, in his will, set the requirements for the suitors who wish to marry her: they must choose between three caskets of lead, silver, and gold. Whoever chooses correctly will win the right to marry Portia. The whole exercise is, in simple terms, a lottery, as Nerissa states in Act 1, scene 2:

...therefore the lottery,
that he hath devised in these three chests of gold,
silver and lead, whereof who chooses his meaning
chooses you, will, no doubt, never be chosen by any
rightly but one who shall rightly love.

Nerissa here also states the belief that destiny is to determine that the person who chooses correctly will not only have made the right decision but will also be the one who truly loves Portia.

Added to this, the will requires that suitors undertake a solemn vow that, if they fail to choose the correct casket, they will never approach another woman for marriage and, therefore, spend the rest of their days as bachelors. Before they are given the opportunity to choose, the suitors have to visit the chapel and undertake the vow in the presence of Portia and other witnesses, as Portia informs the Moroccan prince in Act 2, scene 1:

You must take your chance,
And either not attempt to choose at all
Or swear before you choose, if you choose wrong
Never to speak to lady afterward
In way of marriage: therefore be advised.

Breaking the vow will, of course, mean eternal damnation for the transgressor. If one takes into regard the deep religious convictions held by the general populace at the time, uttering the pledge is a critical step and not one to be easily dismissed. 

This, in itself, is probably the greatest test for all the suitors. If they are successful, they will benefit grandly, for not only will they win the hand of a beautiful and intelligent woman, but they will also own half of the enormous wealth her father has left her. If they fail, though, they are bound to the vow. There is thus an enormous risk involved in the decision to choose.

To complicate matters even further, the will forbids Portia to provide any form of assistance to the suitors. She undertook a vow to this effect and should she transgress, she will be disowned. She has no choice in the matter and will have to accept the outcome, whether she likes it or not, as she states in Act 1, scene 2:

...O me, the word 'choose!' I may
neither choose whom I would nor refuse whom I
dislike; so is the will of a living daughter curbed
by the will of a dead father...

Portia is obviously enormously relieved when Bassanio, the one she loves, eventually makes the right choice, for she had no desire to marry any of the other suitors. 

How do the caskets in The Merchant of Venice test character?

In Act I scene 2 Nerissa makes explicit the reason why Portia's father devised such a strange way of selecting posthumously the suitor who will marry his daughter. Because he was such a "virtuous" and "holy" man, the lottery will only be solved by a man who is able to be "rightly loved" by Portia, because he will display sufficient wisdom, intelligence and humility in selecting the right casket as to make him a good partner for Portia. This is something that Bassanio achieves in Act III scene 2, when he correctly chooses the lead casket and reads the following scroll left by Portia's father:

You that choose not by the view

Chance as fair, and choose as true:

Since this fortune falls to you,

Be content, and seek no new.

The test of the caskets thus discerns whether the suitor is swayed by outward appearances and impressions of wealth, as hinted at by the gold and silver caskets, or whether they are able to "choose not by the view" and select an apparently worthless substance that holds great wealth inside. Because Portia is "richly left," her father wanted to protect her from being the prey of fortune seekers who would only be after her money. Bassanio arguably shows that he is not after money through his willingness to select the worthless lead casket, showing himself to be a true suitor.

Who designed the casket test in The Merchant of Venice?

In the play, Portia's father designs the casket test in his will as a way to exert influence on the choice of Portia's suitor even after his death. It is designed so that Portia will marry the suitor who values her for herself.

Shakespeare took the story of the the three caskets from a well-known folk tale. Other writers who had used it before him were John Gower in Confessio amantis, Boccaccio in The Decameron, and the anonymous author of Gesta Romanorum.

Shakespeare combined at least two folk-tale motifs with details from several other sources to create his play.