What is the relationship between Hero and Leoneto?

Leoneto and "his short daughter" Hero are quite the dynamic duo. In a medieval sense. While "Much Ado About Nothing" is a comedy, it is a comedy of characters stereotypical for its' time.
Hero is demure and innocent, you feel her vitality throughout the play, but her character is left to the readers imagination. You sense she is important, but in the it is her deference that gives her value, not her assertiveness. Hero is painted as the ideal medieval woman. Because of her modesty, other male players can project their idealizations upon her. She is a mystery, a mysterious void for any kind of male fantasy. To her father she is the devoted daughter. To her suitor, Claudio, she is a virgin bride who owes him absolute loyalty. Each man expects her to play the role the project upon her. Her temperament leans toward obedience, and so she seems happy in this role. She loves her father, and though she is an only child, she shows no sign of being spoilt. She represents innocence, and is easy to feel affection for.
Leoneto is gregarious, humorous and generous soul, who frequently wears his heart on his sleeve. He prefers action to thought, which he believes overcomplicates things. He deeply loves his daughter, and one gets the impression he withholds nothing from her. He keeps her in his council, entrusting her into a conspiracy wherein several players trick Beatrice and Benedick, sworn enemies, into falling in love. But there is no question of her deference to him. When Leoneto thinks Don Pedro will propose to his child, he plainly tells her she knows what her answer will be. It was consider the mark of a lady to defer to her father, and the mark of a wise father to know what was best for his daughter. In this Leoneto is an ideal medieval man. Strong and authoritative, but generous and benevolent, he rules his household with an open heart.
While Hero and Leoneto are playing their tricks on Beatrice and Benedick, another and more serious trick is played on them by Don John. Don John makes it appear as though Hero lost her virginity to another man. This presents the first real conflict between father and daughter. When Hero was falsely accused at her wedding, Leaneto became deeply enraged and threatened to kill her. Others intervened, intuiting mischief.
Upon this intervention Leoneto says he will kill Hero himself if the accusations proves true. But if another falsely accused her he would kill him instead. In this you see his character. It seems cruel for our time to murder a daughter for losing her virginity. For the times however, it was not unusual. He also demonstrated her extreme love in his willing to kill on her behalf. He can not quite bring himself to believe the accusations against her. So he tempers his fiery nature and decides to wait for true proof, devising another trick to help bring it to light.
After Hero is proven innocent, the father and daughter team have one last game to play. Convincing her groom, Claudio, that she died by his slander, they guilt him into marrying another in her stead. He gladly accepts, humbled by his mistake. When he lifts her wedding veil, to his joy he finds his one true Hero. Don Pedro expresses his wonder that:
"the Hero that was dead now lives!"
To which Leoneto responds:
""She died my Lord, but whilst her slander lived".
Truly, there was much made ado about nothing.

Hero and Leonato enjoy a close father-daughter relationship. She is characterized as a sweet-natured young woman and an obedient daughter to her father. This depiction of her character is what makes the charge of infidelity against her so difficut for Leonato to bear. The fact that Leonato believes the accusation at first reveals that their father-daughter closeness does not always win out when traditional gender roles are at play. His heartache does, however, reflect the sincere affection and love he feels for Hero.
In modern terminology, Leonato is a single father to Hero, which means that they are closer than perhaps a father and a daughter in a traditional family set-up would suggest. They share characteristics like kindness and gentleness, which reflects the influence they have had on each other and the parenting style Leonato has shown Hero while raising her on his own.

The relationship between Hero and Leonato is a close one. Hero is Leonato's only child; not surprisingly, she is his pride and joy. As with any father of the time, Leonato is very much a creature of the society in which he lives. During the 16th century, fathers were expected to rule their families with a rod of iron, exercising absolute control over their wives and children. And there are moments in the play when Leonato does indeed act out the role expected of him. For example, Leonato instructs his daughter to respond positively to Don Pedro's overtures of courtship:

"Daughter, remember what I told you.  If the Prince do solicit you in that kind, you know your answer." (Act II Scene I)

Yet Leonato is slightly different in some respects to most fathers of the time. For one thing, he respects Hero's judgement enough to allow her to decide whether or not she'll marry Claudio. But there are limits as to how far Leonato departs from the norm. When Claudio stands up Hero at the altar, accusing her of being unfaithful, Leonato's reaction is fairly typical of a father at that time:

"O Fate! Take not away thy heavy hand!
Death is the fairest cover for her shame
That may be wished for." (Act IV Scene I)

Despite his undoubted love for his daughter, Leonato automatically assumes that there's no smoke without fire and that she must be guilty of inconstancy. He subscribes completely to the commonly-held notion that women must be virtuous and not bring shame upon their families, and that the only way to remove the stain of shame in this particular case is if Hero should die.
 
Hero is not a particularly dynamic character. She doesn't make things happen; things happen to her. And she accepts all of this without much in the way of protest, playing the part of a dutiful daughter to perfection. The contrast with Beatrice couldn't be more stark. In that sense she, like her father, can be seen very much as a product of her time. And though Hero's relationship with Leonato is genuinely loving and affectionate, it's also deeply conventional.