I'm currently doing a job of designing a website.
Here, can I say like this?
Designing a website is what I spend most of my time at.
Designing a website is what I spend most of my time doing.
which one is more proper?
I'm currently doing a job of designing a website.
Here, can I say like this?
Designing a website is what I spend most of my time at.
Designing a website is what I spend most of my time doing.
which one is more proper?
Ideally you would reverse the construction to remove the extra clause. Something like
I spend most of my time designing websites
However, you could use a construction like yours for emphasis. Imagine this conversation:
"You should build experience before you look for a job. Try designing a website."
"But designing a website is what I spend most of my time doing!"
It's a little awkward, but it works. Either way, I would avoid your first construction, simply because it needlessly ends a sentence with a preposition. This is a rule that is often broken in American English, but if it's easy to avoid, do so.
As Winston Churchill probably did not actually say:
Ending a sentence with a preposition is something up with which I will not put.
As Ahmad pointed out in his answer and in the comments, it would be more common to use "designing websites" here, to say that you are generally designing an unspecified number of websites. to say that you spend most of your time "designing a website" means that there is a single website that you work on and it consumes a lot of your time. If that is what you mean, it would be clearer to specify which website.
When you make such relative clauses, you should be able to read them in reverse
I spend most of my time at designing a website
It sounds grammatical, but we don't use an activity as the adjunct of spending time, however you could say This park is where I spend most of my time in
Second example:
I spend most of my time doing designing a website
However it's awkward but it still is grammatical, because designing a website is a noun phrase like doing something, but we don't say doing + verb + ing except for a discrete activity (Watching, Drinking or Designing websites). we also say doing exercise, doing homework, then you could say:
Exercise is what I spend most of my time doing
Designing websites is what I spend most of my time doing.
Drinking is what I spend most of my time doing
But they are again awkward and just for emphasis purpose, it is natural to say them as
I spend most of my time doing exercise
I spend most of my time drinking
I spend most of my time designing websites