One, scene one, Henry's opening speech relays information to the audience that war within the borders of England has him greatly concerned—while he is making plans to partake in fighting in the Holy Lands, he realizes that he must direct his attention to difficulties at home first.

In Westmoreland's report to the King, we learn that Glendower, the "Welsh rebel," (fighting alongside the Scots) has taken the loyal Mortimer and murdered a thousand of his men; in doing so, the bodies of the dead have also been mutilated—it is so awful that Westmoreland cannot even speak of it. Glendower is portrayed (with this news) as barbaric, and the leader of uncivilized people.

…the noble Mortimer,


Leading the men of Herefordshire to fight


Against th' irregular and wild Glendower,


Was by the rude hands of that Welshman taken;


A thousand of his people butchered,


Upon whose dead corpse' there was such misuse (I.i.37-42)

Henry decides that the war in the Holy Lands must wait. Westmoreland then reports on the success of  "the gallant" Hotspur and those with him...

...At Holmedon met... (54)

Hotspur (and a Scot that serves the King, Archibald..."approved") has, on another front, fought and defeated "the Douglas" and his enormous army. The King has already had reports of this (from Sir Walter Blunt) and is greatly pleased by Hotspur's success. As the King relays details of Hotspur's defeat of the Douglas and the taking of a number of notable prisoners, Henry is delighted at Hotspur's accomplishments:

And is not this an honourable spoil,


A gallant prize? ha, cousin, is it not? (73-74)

The King envies Hotspur, the son of Northumberland:

Should be the father to so blest a son,--


A son who is the theme of honour's tongue;


Amongst a grove, the very straightest plant;


Who is sweet Fortune's minion and her pride... (78-81)

While Hotspur does his own father proud, and good fortune smiles upon him in battle, showing him to be admirable and praiseworthy, the King feels quite differently about his own son. He laments the shortcomings of Harry (Hal). The depth of his disappointment is evident. Even while he praises Hotspur, he wishes that the fairies had switched the children at birth so Henry could claim Percy (Hotspur), while Hal (the Plantagenet) would be seen as Northumberland's son. Henry's disappointment in Hal is profound:

See riot and dishonour stain the brow


Of my young Harry. O, that it could be proved


That some night-tripping fairy had exchanged


In cradle-clothes our children where they lay,


And call'd mine Percy, his Plantagenet! (83-87)

Hal spends his time not fighting for "King and country," but hanging about with Falstaff, the corrupt (yet comic) leader of a gang of "ruffians." While Hal enjoys his time with them, he knows it is temporary. Eventually, Hal will turn himself around, becoming a man worth admiring.

Most of the characterizations we get early on in the play come from the King's discussion of fighting with the Welsh and Scots, and the effort (or lack) of Henry's men on his behalf, with regard to the war raging in England.

One of the fascinating elements of the History plays is the way in which they intersect with actual history. It is key to have some knowledge of the rather messy and bloody history of the crown in England in order to answer your question. Let us remember that King Henry IV gained the throne through rather nefarious means: he deposed and murdered the king that came before him, Richard II. Therefore, when the play opens, we see a man who appears to be wracked by guilt and upon whom the crown sits very heavily. Note the opening words of the play and how they emphasise this impression of his character:

So shaken as we are, so wan with care,

Find we a time for frighted peace to pant

And breathe short-winded accents of new broils

To be commenced in strands afar remote.

Note how Henry describes himself as being "shaken" and "wan" with the responsibilities of the throne. The alliteration of the "p" sound in the second line emphasises the way that the current peace he enjoys is only viewed by him as a chance to swiftly draw breath before the "new broils" descend upon him. He is therefore a man who, having achieved what he wanted, appears to be weighted down with guilt and with the realities of being king. Note how he mentions the desire to go on a pilgrimage to atone for his rather bloody assumption of the crown.