The Easter Duty is to go to Confession and receive Holy Communion at least once per year.  The duty is to be performed during the Easter Season, unless there is a compelling and sufficient reason to perform it at another time during the year.  The Easter Season is the period from Easter Sunday through Pentecost.

It can be fulfilled from the First Sunday of Lent to Trinity Sunday (the eighth Sunday after Easter).

After a bit of research, the current (1983) Code of Canon Law defines the time for fulfillment of the Easter duty as from Palm Sunday to Pentecost Sunday.  In the United States (and perhaps other countries) that has been extended as the time from the First Sunday of Lent to Trinity Sunday (the Sunday after Pentecost).

The 1917 Code of Canon Law says:
 

702. Each and every Catholic of either sex must, after having attained the years of discretion, receive holy Com munion once a year at least in the Easter season, unless at the direction of his priest he believes it proper to abstain for some time.

 The Easter Communion is to be received between Palm Sunday and Low Sunday; the local Ordinaries have the right to extend this period for all the faithful of the diocese, if they think this necessary on account of the condition of people and place, from the fourth Sunday in Lent to Trinity Sunday, but no further.

The faithful are to be advised to satisfy their Easter duty in their own parish church, and those who received in another church should take care to inform their pastor of the fulfilment of their Easter duty.

 If for any reason one has not made his Easter duty within the prescribed time he is still bound by that precept. (Canon 859.)

By my reading, according to both the 1917 and the 1983 Codes of Canon Law, one who receives Holy Communion at the Easter Vigil will have full filled their Easter Duty.

Confession is not part of the Easter Duty, though many seem to think this.

We must confesses our sins once a year.  This does not have to be during the Easter Season.

The Easter Duty requires we receive Holy Communion worthily once during the Easter Season. 

But that also means that if someone has mortal sin on their soul they must also go to confession beforehand. 

If somebody receives Communion on Holy Saturday during an Easter Vigil does he fulfill his Easter Duty or does he have to receive on Easter Sunday or after to fulfill it?

Easter Vigil (the part where you receive Holy Communion) IS considered to be on Easter Sunday.  I'm guessing you're at one of those parishes where the Easter Vigil (including Easter Mass) is celebrated on Saturday morning (a practice I dislike), but according to those rules, they hold that the Lent is over with the Easter Mass of the Vigil.  There's a bit of ambiguity in terms of whether one believes that the changes made by Pope Pius XII to change the time were legitimate.  I hold that they were, especially the time aspect.  If one were living in 1957, would that have satisfied your Easter Duty if some priest still did the Vigil on Holy Saturday Morning?  Not sure about that one.  I suspect that's where your question is coming from.

But why not just receive Holy Communion again on Easter Sunday just to be sure, or at least at sometime during the Easter season afterwards?

I don't know the history of changes on this rule, but the current rule is First Sunday in Lent -- until -- Trinity Sunday.

Right.  There's a bit of confusion.  I believe that even before Vatican II, the US bishops at least allowed making the Easter obligation beginning with Lent (though Trinity Sunday, you say).