The first word in these lines, "but," is a word that means that what follows is in contradiction of, or in contrast to, or in explanation of what has gone before. When used after a negative statement, "but" can also mean that what follows is actually the truth of the matter.

As far as analyzing these lines is concerned, "but" means that these lines have no meaning outside the context in which they appear.

So let's look at the context:

QUEEN: Good Hamlet, cast thy nighted color off . . .

"Quit being so gloomy."

And let thine eye look like a friend on Denmark.

"Make friends with Claudius, and look kindly on me, even though you might not like that I married him, and so soon after your father died."

Do not for ever with thy vailed lids
Seek for thy noble father in the dust.

"Buck up."

Thou know'st 'tis common. All that lives must die,
Passing through nature to eternity.

"Everybody dies."

HAMLET: Ay, madam, it is common.

Everything that Hamlet has said so far has been a play on words, and this is no exception. He's saying, "Yes, it is common that people die," and he's also saying, "Yes, it is common that people act sad about it," and, "It is common that you should say that."

QUEEN: If it be,
Why seems it so particular with thee?

"We've had a nice wedding, great reception, good music, dancing, lots of terrific gifts. We're all dressed up and having fun, and you're moping around the place like somebody died. So what's your problem?"

HAMLET: Seems, madam? Nay, it is. I know not seems . . .

This is the kind of line that an actor can really play with . . . "Seeems, madam? Nay, it iiiiis. I know not . . . seeeeeems." Overacting, sure, but the point is that this line—in fact, just the word "seems"—encapsulates one of the major themes of the play: appearance versus reality. Hamlet is all about "reality," and he hates "appearance."

Up to now, Hamlet has said only three lines—all short, clever, cryptic, and sarcastic—and he could have stopped talking, but this is something that Hamlet feels deeply about, and he's going to say so.

. . . 'tis not alone my inky cloak, good mother,
Nor customary suits of solemn black,
Nor windy suspiration of forced breath,
No, nor the fruitful river in the eye,
Nor the dejected havior of the visage,
Together with all forms, modes, shapes of grief,
That can denote me truly. These indeed seem . . .

That word again. Shakespeare really knows how to do this.

For they are actions that a man might play; . . .

"Don't be deceived into thinking that because I'm walking around here in my black clothes, looking sad, and crying my eyes out that I'm just acting all of this."

And here it comes . . .

But I have that within which passeth show,
These but the trappings and the suits of woe. (1.2.70–89)

This is the first rhymed couplet in the play—something that Shakespeare frequently uses to end a scene. The scene could end right now, but Shakespeare uses the couplet here for emphasis. What Hamlet says is something that Shakespeare wants the audience to pay particular attention to, and remember.

Hamlet's not just saying that he's actually sadder than he looks. This is ominous.

Hamlet has a lot of things on his mind right now. His father died under what looks like suspicious circumstances. (He was bitten by a snake while he was taking a nap in the orchard behind the castle? Not bloody likely). His mother married his uncle about two minutes after his father died, which leads Hamlet to believe that the "incestuous" thing between them has been going on for quite a while. His uncle usurped Hamlet's rightful place on the throne—which is something that nobody talks about much, but it has to be in the back of his mind. This is all building up inside him, and like Hamlet says, something is going to come out, sooner or later.

It's going to take about four hours (it's a long play) for whatever is churning around inside Hamlet to come out, but when it does, there are going to be a whole lot of dead people laying around the stage, including Hamlet.

And his mother wants to know what his problem is? She has no idea.

Unlike everyone else in attendance at this important assembly and celebration, Hamlet is dressed in black mourning. Both Claudius and Gertrude are trying to persuade him to stop mourning for his dead father, which would mean giving up his black clothing. Hamlet tells his mother

'Tis not alone my inky cloak, good mother,
Nor customary suits of solemn black
.............................................
That can denote me truly
...........................................
But I have that within which passeth show.
These but the trappings and the suits of woe.

In other words, his mourning clothes are only customary and obligatory symbols of feelings of grief, but he has real feelings of grief for his dead father which he cannot show or express. One reason he cannot express them is that they are involved with what he considers his mother's adulturous and incestuous union with the new king.

Both Claudius and Gertrude seem unable to understand why Hamlet should be experiencing such strong emotions of grief for such a long time after the event. Claudius suspects that Hamlet has other reasons for being so depressed and withdrawn besides mourning for his father. Claudius will spend the rest of the play trying to figure out what is going on inside his stepson's heart and mind. He is pretty sure that Hamlet is harboring bitter resentment at having the crown snatched away while he was still at Wittenberg, and he can tolerate that resentment as long as his stepson isn't going a step further and plotting against him. That is what he wants to find out. Gertrude only thinks her son is unhappy because of his father's death and what she calls her "o'erhasty marriage." She is not suspicious of her son's political intentions, and at this point he probably has no thoughts of usurping Claudius.. He only wants to go back to school at Wittenberg, but Claudius refuses him permission because he wants to keep him where he can watch him closely and have others watch him closely as well. At this point Hamlet has not met with his father's ghost.