If I design the arduino uno R3 kit by myself, does I need any firmware or it suppose to work as the original kit?
if yes, please inform me with the convenient links
(deleted)
You can buy the ATmega328 chips either with or without the bootloader already installed. In general, having it pre-programmed will cost more.
The 32U4 chip pre-programmed will be a bit more difficult.
The two big questions are why you would ever want to spend a lot of time, effort and money to replicate an Arduino when they are so cheap as clones - even actual UNO clones - on eBay and why you would want to use the inconvenient form factor of a UNO? 
I don't understand your question...
If I design the arduino uno R3 kit by myself
What do you mean by "kit"? There are "starter kits", or "educational kits," or "experimental kits" that come with an Arduino plus some other parts/components. Yes. You make your own kit.
.
does I need any firmware or it suppose to work as the original kit?
The only firmware it comes loaded with is the bootloader. If you buy an Arduino it comes with it. If you are making your own Arduino board "from scratch", you either buy a chip with the bootloader loaded, or you can program it yourself. Without the bootloader you don't have an "Arduino". Typically you use the bootloader to load your own custom firmware, but of course you can load the examples or other software written by someone else.
With the right hardware you can load the bootloader yourself and you can find instructions on the Website. That might be worthwhile if you are selling/distributing the kit, but for your own use if you can program the bootloader that means you can load firmware without a bootloader so you don't need a bootloader. 
DVDdoug:
The only firmware it comes loaded with is the bootloader.
Actually, a real "UNO" or clone thereof, has the 32U4 as well, which has the USB-serial firmware as well as the bootloader on the 328.
Now of course, many "UNO"s are fakes, actually being a variant of the older Duemilanove design (CH340 instead of repudiated FTDI chip) with a couple of the extra headers like the UNO. They are necessarily not UNOs as they lack the functionality of the 32U4 - not that you often will need to use it. 
But again, constructing a UNO from scratch for your own use, let alone proposing to market it against the clone manufacturers, is the most absurd exercise in futility imaginable!
Follow this tutorial to make a "bare bones" Arduino that will do everything the Uno does!