How do I remove this kitchen sink?

Try to cut any sealant at the edges. Most sinks have some type of bracket that holds sink to counter bottom. Remove garbage disposal and drain pipes first. You will need help holding sink to keep it from dropping on your body.

You may end up taking the smaller tiles around the sink off. Normally there's a flange on the sink and I would bet those smaller tiles are covering it up. If you take them off carefully you should be able to reuse them.

Yes, but! This does not get the sink out from under the tile around the edges. Not sure this is a DIY.

grout saw to cut/remove grout without damage to tiles yes it is DIY just becareful this sink may be castiron under ceramic/porcelain coating & will be real heavy. post online,craigs list for resale or donate to Habitat for humanity some one just may need your old sink. "one man's trash is another man's treasure. Or you could keep as laundry room,or potting shed addition. or even repurpose as fountain or planter.

Under-mount sinks are usually held in place by caulk (shouldn't be grouted!) and sink clips. This should be a two-person job. First shut off the water supply to the sink, then disconnect the sink from the supply pipes and drain, put a bucket under the sink to catch any water (you may want to put something sturdy under the bucket to raise the sink so it doesn't fall into the cabinet during the next steps), use a box knife to cut out the caulk, then remove the clips. It may be a little stubborn at first but a little wiggling and pushing should free it from the opening. I hope this helps!

Not being able to actually see how the sink is situated in the opening is an issue. As this is an old sink and it looks like they just tiled over the existing countertop it probably has a lip around the opening so you will need to remove the edge trim tile to confirm that it does have the over the counter lip.

Remove grout from tiles and the carefully remove the tiles. That will release the top of the sink.

Then go underneath the sink and disconnect plumbing. While your at it check out the installment clips that are used to tighten and hold the sink in position. These will need to be removed but as mentioned by others if your sink doesn't have an above counter lip you will need someone to hold the sink up so it doesn't drop on you. If it does have the above counter lip it should hold in place until you pick the sink up from the top.

You don't state whether you are replacing it with a top mount or bottom mount sink so you will need to look at the opening and see if which would be best. By the looks of it in the picture you will have quite a big drop off from the counter top that you may have to build back up so that your new sink is flush with the countertop.

Because of that you may want to call in a professional because while installing a new sink isn't normally that hard I can see the issue with the depth of the tile being one.