I have a Squier Deryck Whibley Telecaster that I've had about 6 years. Here's a pic of it:

It's an alright guitar, it gets an ok sound from the one pickup. But I really think that it'd be better if it had a neck pickup. I've been wanting to do this mod for a while to this guitar, but the problem is this:

It's completely solid wood where the pickup and toggle switch would go. The pick guard's the same way too.

So what I'm wondering is would it even be possible to install a lipstick neck pickup and toggle switch into this guitar? And if it is possible, how would I go about doing it?
Yes it is possible.

Mark out the area you want to remove and use a drill with a brad-point (forstner) bit and carefully drill out the majority of the area. Then use a small chisel to remove the rest of the material.

You can also use a router or roto zip if you have them (it works better). You then wire a switch and mount it and your in there
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If you do the chisel thing, you may want to read up on how to use a chisel. It may seem obvious but it does matter which way the bevel faces depending on what you are doing.
This is a relative easy thing to do on this type of guitar (assuming some basic tool use knowledge!) because any mistakes are covered by the scratch plate any way however, easier and possibly better options may apply with relatively cheap wiring alterations?

1. Coil split the humbucker so it becomes a singlecoil to give more tonal variety (this may require a new pickup)

2. Fit a second volume control and a switch to allow you to easily switch between a clean/dirty sound by rolling the volume back (which I suspect is your issue?)

There are tons of options which you can do without butchering the body itself, but to be fair it sounds like you've simply outgrown this guitar and should think about upgrading rather than spending good money trying to get a better sound out of fenders driftwood models.

They are good guitars in their own right but..... yeah swap it altogether is my advice, then you still have this one unaltered too