Right now it's thrip season, and being next to a barley field doesn't help much for keeping your distance from them. This thrip must have decided eSheep was too outdated, and needed an upgrade, so I left him trailing around: up and down my screen for a while, until the temptation to squash it set it, causing an effect not unlike 5 dead pixels all together... Great.

-_{thanks to HowSuffWorks for the image below.}

Step 1: A Word of Warning...

You might want to Google for thrips stuck in monitors, it's not that unusual, Here's a link to Dell forums about someone with the same problem. If you also have a spare monitor around I suggest taking it apart for practice, decreasing the likeliness of screwing up when it matters.
As mentioned in forum (above link)
The unit would have to be disassembled to clean it out. That is a job for a professional.
Have you checked with a PC repair shop to see if they can do that (and guarantee their work in case they mess it up).

-I've already taken laptop screens apart, so I had a rough idea of what to do. ;-)

-If you want to attempt this, having taken apart electronics &c. is recommended. Please Note: you may be able to get a replacement if your monitor/laptop is under warranty.

Step 2: Tools for the Eating...

You're going to need some basic tools:
I used:

Flat head screwdriver (for prising apart the monitor casing)
Phillips screwdriver, smaller size (for taking out all the screws)
Precision screwdriver (for all the smaller screws)

Window Cleaner* (you do remember this is to remove a thrip)
Micro-fibre cloth (any cloth will work, but micro-fibre cloths are better)

*You might want to check out this Instructable (although IMHO it's a little OTT =) )

Footnote: This is my excuse for the varying quality of the pictures, 'tis a several year old camera, which I am quite un-familiar with.

Step 3: Remove...

Just so you know, laptops all vary, so you may have screws, cables, where I don't. &c.

First thing to do is to remove all power to the laptop/monitor, Power cable, Battery...you know what they are. The monitor back-light uses high-voltage electricity (so the warning says) stay safe.
Next you'll need to remove the plastic casing around the screen, On mine there are six screws with rubber "feet" on top, these can be removed with a fingernail. (Pictures 2&3)

Un-screw all the screws, a put them somewhere handy, either remember that they are for the plastic casing, or use a note of paper! (Picture 4)

The casing is most likely snapped together, Front with the Back. go around it carefully with the flathead screw driver, make sure not to slip and jab the screen. (Pictures 5&6)

After it is un-snapped the front casing will come off, the actual LCD panel (monitor) is attached to the hinges, and the hinges to the back casing. (different laptop makes may differ) (Last picture)

Step 4: Break Away...

Now you should have something like the picture below:
The next part is to remove the rear plastic casing, mine was screwed to the hinge supports. (Picture 2, 3,&4 )

Basically you want to have the LCD screen totally separate from everything, do what it takes, you're not in great danger of messing up here.

-_{Thanks to this site for the last picture.}

Step 5: Down to the Core...

Now for the actual dismantling of the screen, and the extermination of the thrip.

Objectif: Take off the metal bezel(suround).

On mine there was "tape" securing the metal bezel, just a simple matter of peeling it off. you will need to unfold* PCB board that will be on the back. (Picture 3) unfold it, taking care as it is most likely quite thin, and flexible, you will need to remove the white tape too.

*_{It's just folded around, onto the back of the LCD panel, it's not necessary to remove it, and removing may lead to unfortunate circumstances.}

(Picture 4)
After the metal bezel is off the panel will most likely just split into two parts, the pixel matrix layer (on the top) and the back of the LCD panel with other white plastic layers, don't touch these unless your problem is not on the pixel matrix layer!
...Now it is just a case of cleaning the back of the pixel matrix layer... ordinary window cleaning works very well, or you can go the longer route of this Instructable.

Step 6: ...Reversi

Now that you've taken it apart, you should know who to put it back together. :)
...or just follow the Instructable backwards. =)

Replace the pixel matrix layer.
Put on the metal bezel.
Stick the tape back on
Screw it back into the hinge supports, make sure all cables are connected, then screw it nto the rear monitor casing.
Snap on the front plastic, and screw in, replace rubber feet.
Plug the monitor/power back in, and test.

-That, I believe is pretty much it. :-)