How to remove grating from bathroom sink
The grating is likely part of a threaded barrel that sleeves through the porcelain of the sink and comes out the bottom. This barrel has threads on it under the sink, and a wide and thin nut is threaded onto them squishing the barrel and grating against the sink and securing it. 

To remove it, disconnect the P-trap beneath the sink first (have a bucket handy) and set it aside. Then find this wide thin nut up against the bottom of the sink and unscrew it. You'll probably need a big adjustable wrench or a pair of "channel locks" AKA "water pump pliers" to remove this nut. Also look for any drain closing mechanism that may be present here and connected to the tube and grating you're trying to remove. There are different styles of this mechanism so just take it slow and assess for yourself how best to disconnect it - it's usually just a matter of a finger turned nut or a small metal clip to be removed to free the connection.

When you do finally remove the grating and barrel you'll find a soft waxy substance beneath the rim of the grating between it and the porcelain of the sink. This is plumber's putty and seals this joint. You'll want some fresh stuff on hand when you go to put everything back together (it's cheap and widely available). You simply make a small rope of the putty and lay it in there using more than you'll need (after getting the old stuff out of there and cleaning the joint well!), and tighten the nut down squishing out the extra putty which is easily wiped away. 

And maybe you'll get lucky and the clog is just in the P-trap, or you can clear the clog without taking the grating and tube apart from the sink if it is in there once the P-trap is out of the way. And hopefully the clog is present somewhere close to the sink here and not further down the line!