By what means does Shakespeare build suspense before the Ghost's appearances?What disturbing political events occur in the background of the first...

These are two distinct questions, though both have to do with the introduction and setting of the play and are found primarily in the first scene.

First, suspense.  Several things which happen at the beginning of the play set the scene for something rather ominous to come--most notably, the appearance of the Ghost.  Look at the first words spoken in the play.  "Who's there?" is what the man on watch (Fernando) should have spoken to the man who approached him; instead, it is Bernardo who asks this question as he approaches the guard.  Something's not quite right even from these opening lines (though it is easier to catch this visually rather than in the reading).  We quickly find out it's midnight, bitterly cold, and so still not even a mouse was stirring--all signs that trouble may be ahead.  Their conversation turns to the appearance of a ghost, and the reasonable voice of Horatio makes us believe there is nothing to all this talk, even though the Ghost has been seen on two other occasions...and then the Ghost appears.  Even Horatio is shaken by the sight, and the audience is hooked.

Now the disturbing political events.  First, the Ghost marches toward them in full military array, a sure sign there is trouble on the horizon for Denmark.  Next, it is the ghost of the recently dead king, Hamlet, wandering restlessly in his kingdom--another indicator that all is not well in the country.  Horatio says, "This bodes some strange eruption to our state."  This prompts Marcellus to ask why there is suddenly such a strict patrolling of the land, why there has been a buildup of munitions, and why shipwrights are working around the clock making ships.

Horatio explains that the former King Hamlet had defeated Fortinbras of Norway (in a duel provoked by the hot-headed Norwegian) and as the winner had legally taken possession of Fortinbras' lands as decreed in a signed, legal document.  The son, young Fortinbras, was also quick to fight and had gathered an army of mercenaries to fight with him to regain the lands his father had lost.  Denmark was preparing for that imminent battle.

This occurs in the first scene of the play; the politics briefly continue in scene two. The new king, Claudius (brother of old King Hamlet and new husband to his former sister-in-law), sends messengers to Fortinbras' uninformed uncle to tell him of his nephew's plans.  He hopes to thwart the impending battle before it reaches his doorstep.

The family dynamics (politics) have just begun; however, this is the gist of the more national political and martial unrest found in Act One.  The trouble is obvious enough that one of the relatively uninformed guards speaks the famous line in scene four:  "Something is rotten in the state of Denmark."