Bringing Computer Tower & Monitor to Vietnam

It should not be a problem as long as the packaging shows clear evidence that the computer is your used personal property.  Even if you incur excess baggage charges, it might be best if it had it's own box so that they can't claim that you are trying to hide it under your clothing.  I moved a 27 inch iMac to HCMC.  When I got there the customs people asked "new or used" and passed it right through when I said used.  It was in the original box but had clearly been taped and re-taped more than once.  On the way out they tried to shake me down but I held firm.  Of course they were displaying their own ignorance as it is ridiculous to suggest that someone would bring a new iMac to the US from Vietnam where it costs about 20% more.

Are you removing the internal hard drive?  Seems like a lot of trouble.   Why not just clone it to a portable and keep that in your carry-on?

Its a gaming beast. There's no hiding this.  We're going to downsize the tower though so it meets carry on requirements

When ever I travel I bring a laptop and a 24 inch monitor with me.

The monitor (photography grade) is packed in the original packing (OK the outside has more tape on it due to wear and tear but the internal packaging is excellent) and goes into cargo.

The laptop comes onboard, along with several external hard drives as carry on.

As the most important equipment I travel are with me in the cabin (camera, lenses and computer) I feel secure in the knowledge that they are safe.

I have never had any challenges with this arrangement either entering Vietnam or Thailand.

My advice is to leave the tower home and make travel life a lot easier with a laptop.

Horses for courses.
A laptop is always going to be easier, but I know some people really need a massively powerful PC.
My personal daily carry is far from a problem as, except for three programs I don't use all that much, all I need is a Samsung tablet with keyboard case.
Gamers will tell me it's a waste of time, but I'm old, fat, ugly, and uncool, so I don't play such things. If I need a big screen, I use a 32 inch Samsung TV - It works a treat.
Using a locally bought TV might very well solve at least half of the OP's issue, but he'll know far more about his monitor requirements than I do.
Still, with the shockingly cheap price of a decent TV, it might be an option.

just remove the motherboard hard drive and leave the rest, when you get here buy another tower and refit. https://www.youtube.com/watch?v=UM-9wAUQCdU  good idea to bring ur power supply and DEfinately label ALL connections before removal