Would I have to patent my computer systems (such as desktops and laptops) in order to produce them and sell them to markets? Similar to Dell and HP would operate.
Unless you are creating a new type of computer (not a desktop or laptop), I don't think that applies.
If your method of creating a computer is truly original, then maybe you could receive a patent. Most likely if you are just putting together computer components (like Dell and and HP), you aren't producing anything that is patentable.
I don't know all the details of your business, but from your brief description, it doesn't sound like you have anything that is patentable. Producing everyday computers in the same way Dell and HP do is not unique (because these companies are already doing it), and as a result you will not be granted a patent for that. You must be able to show something unique (that hasn't been done before) in your process or product in order to be granted a patent.
But to answer your questions:
The answer is no. You do not need a patent in order to produce computers. In fact, you will never actually need a patent to take anything to market. A patent is simply a way to deter others from copying your product/idea, but it is not required for any business venture. The only time you may "need" a patent is if you are looking for VC funding. VCs like to see patents.
That said, both Dell and HP hold lots of patents. If you are planning on going into business as a computer manufacturer, you need to become familiar with their patents, and make sure you are not infringing on their patents. See the links I provided for a list of their patents. You will need a lawyer to help you with this.