Also, i may plan to plant some climbing plants in the planter so that they can climb and shade our patio. But what's the dimension of the planter i may need?

And any suggestion for the climbing plants?

I don't think it would last a season.
Posts set in a planter will not support even a trellis type roof through a storm, much less a trellis covered in vines.

At least not where I live.

Yep- that would be horribly unstable.
Setting posts in concrete is not a difficult thing to do- I'm a little old lady and I can do it. Digging the holes is the hardest part but it is a simple enough task.

This structure is going to be permanent so do it correctly- otherwise you are going to have a falling down eyesore in your back yard instead of the picturesque cover of vines you are envisioning.

Thank goodness for code compliance and the permit process.

Yes, you require a footing for an overhead patio shade structure.

Most likely you will need to submit a plan to your building department to receive a permit to build the structure, so someone with experience with this is required. A contractor or landscape architect or even yourself can do this if you have the knowledge .

The dimensions of the planter depend on a variety of elements such as your climate (freeze/thaw/heat) and the variety of vine that you choose.

Actually i get the idea from another guy in this forum.

You can see the link:http://forums2.gardenweb.com/forums/load/strucs/msg0621470222226.html?13

So, i think it may work.

Here is a link that might be useful: anchor pergola with a planter

If you read that thread he still used concrete:

"I built 4 large planters and then I got 4 column forms from Home Depot and set the pergola uprights in them and filled up the forms with concrete.

Then, after the concrete had set, we anchored the uprights through the planters as well (as kudzu9 suggested) and then filled all the planters with dirt. "

I don't get the aversion to digging four simple holes.
Surely a couple of hours of work is worth it to know the thing will last.