So this computer i'm planning to get meets the requirements for the base game, but I don't know if it will also work with 5/6 expansions. I have Late Night, Generations, Supernatural, Seasons, University Life and I want to get Island Paradise too.
Anyway the specs:

OS: Windows 7 Home Premium 64 bit
CPU: Intel Core i5-3570 Processor (3.4GHz)
RAM: 4GB 1600 MHz DDR3C (upgradable to 16GB)
GPU: NVIDIA GeForce GT530 (2GB)

So according to the system requirements wiki, the graphics card runs the base game well, the others up until Supernatural on Maybe which is okay-ish, and Seasons on No which means it has known problems or there are lags.
I just want to know if I could run these expansions well though?
I wouldn't mind a bit of lag. Honestly, I just want it to run fine. If it could run everything without Seasons well, then I'd play without Seasons until I could get a better graphics card. So, ?

Well - bear in mind that if it's a No for Seasons, that means it's also a No for UL and IP, so you'd have to play without both of those too. Honestly I'm surprised that that card does so well with the earlier EPs, because the model number suggests it's a pretty poor card. You'll also need more RAM than that, especially with a weak graphics card. Honestly, I would suggest you save up for a bit longer and get more RAM and a good graphics card - something in the top half of the newer ranges, a 650+ or 750+, rather than a two gen old entry level card.

Remember, too, that desktops are always better value for money than laptops.