We bought a house that already has raised beds. One is FULL of grass and weeds, too much and deep to pull by hand. I understand I should/can put newspaper or cardboard down and then mulch, but do I then pull the dead weeds/grass before planting veggies or do I do that right on top?
Once the weeds are completely dead, unless the soil is very soft, you'll usually want to loosen up the top layer. This is easier once the dead layer is removed. And some weeds carry vegetable diseases (even without symptoms) and can infect crops. An example in my area is hedge mustard, a common weed, that almost always carries clubroot where I am, so I can't grow Brassicas in an area where I've seen hedge mustard, for a couple years.
I'd recommend raking (or cutting, if they're tough) the grass/weeds once they're dead, and working up the top layer of soil before planting. If you could work in a little (or a lot, I guess :D) compost in, that's even better.
You can plant through dead material, and get good results, however. I've found, for instance, that broccoli (in my area) likes very firm soil, so I don't work it at all when I am ready to plant the starts. I just pop them in, through the dead plant matter, and then add more (usually as lawn clippings) to make it weed proof. That has worked great for me.
I've found it best to pull out all grass and weeds and replace the soil with well rotted compost and manure. It's a PITA, but this way, you can till the soil and start fresh. Just grab a shovel and dig away. Plus, I've found that grass is impossible to eradicate and the tiniest bit of root can easily take over. Starting fresh will also make it easier to weed and you'll know that all the nutrition you put into it will only be used on the plants (as opposed to any leftover weeds/grass that's still alive). It'll also make aerating the soil much, much easier since you don't have to worry about all the dried-up and tangled roots from the weeds. Good luck!
Putting down the newspaper will block sunlight to the weeds, so they will probably die. You can cut openings in the thick layer of newspaper to plant your new plants. Then they will have the sunlight advantage over the weed competition.