Look into the what the specifications of the TV is too. Response time of the CRT monitor depends, like for example:
https://www.amazon.com/VG248QE-1920x1080-144Hz-Gaming-Monitor/dp/B00B2HH7G0/ has a 1 milisecond response. It can be connected VIA HDMI or DVI. Playing Melee on the Dolphin Emulator with this monitor is a pretty flawless experience in DVI mode. I haven't tried HDMI, but since it's a 144Hz monitor, I assume there will be then be significant input lag.
In comparison, if you have one of those new HDTVs - the larger then screen, the more input lag there is. By default HDTVs have a 60Hz refresh rate, but in my experience when you get these TVs (Insignia TVs for example) they aren't actually rated at 60Hz and performance enhanced for older game titles for older consoles like the Nintendo Gamecube (SSBM). It might also be something to do with the transfer of the video signal. If played on a Nintendo Gamecube, composite AV cables were designed to work with the Nintendo Gamecube game titles. As soon as you play on older consoles that rely mainly a 60Hz constant refresh rate stream, this is the important bit to focus on. HDMI cables doesn't really do this with older gaming consoles if your attempting to achieve this.
If you decide to play Super Smash Bros. Ultimate (Or any newer Smash game after Melee.) it probably won't be too critical to have the refresh rate matter past 60Hz. In Melee mostly (Because it is the fastest action per minute Smash game.) it is important to play on CRTs as the AV cables give them a constant stream of video input. This all depends on your preference though and your mileage may vary on what television hardware you purchase.
Other then this, vanilla Melee plays pretty fluent on a Durabrand CRTs that is 13 - 17" for example with no input lag. They can easily be bought at tech second hand store for very cheap.