Whew - simple question, but not so easy answer. It depends on all sorts of things ...
If the player is operating, any current taken to charge it will (obviously) be in addition to the normal operating current, so overall it will be consuming more.

But if you can charge the battery with the DVD switched off, the current drawn will be entirely dependant on the design of the charging circuit.
Some chargers charge at a constant current, while others charge to a particular voltage, which roughly means that the current gradually reduces as the battery charge builds up, shutting down to a trickle charge once the battery is full. Some intelligent charges do all sorts of clever things designed to keep the battery in tip-top condition. You might get some idea of the charging current by reading the label on your charger. This will give you the maximum current which it can provide - in practice "normal" charging current would be something less than the maximum.
Once apon a time, the recommended charging current was at the "10-hour rate" - which I think roughly means that it would take 10 hours to fully recharge the batteries. If the DVD player were to flatten the fully-charged batteries in say 5 hours, it would be a reasonable rule-of-thumb to guess that the charging current was half the operating current.
Modern fast chargers though will charge at a much higher current than the 10-hour rate, so again it's difficult to say .... But you might be able to use the same principle here. If your DVD player runs for 6 hours on a fully-charged battery, measure how long it then takes to fully recharge the battery (with the player switched off) - and that should give an idea of the difference between operating and charging current.
I guess I would expect that "normally" something would take longer to recharge than it takes to flatten the charged battery, suggesting that the charging current is lower than the operating current. BUT with some fast chargers, batteries can be recharged very quickly - which must be at a higher current.
Just to add to the confusion, the various currents depend very much on the type of battery being used. Lead acid batteries (like your starter and leisure batteries) need a certain current to charge them to a particular voltage. Older electrical appliances tend to use NiCd batteries, but these have pretty much been replaced by NiMH, Lithium ion or LiPoly - all of which require different conditions to charge them correctly.

Also batteries get tired as they get older; a tired battery is more difficult to charge, a lower current passes (less charge goes in) and it goes flat more quickly.
If it helps, you can roughly think of the battery like a bucket of water. You can 'charge' it through a small hose (low current) or a large hose (high current) - and the time it takes to fill depends entirely on the flow rate through the hose. Usually you can pour the water out of the bucket quite quickly, and it then takes a long time to refill. However, if you're pouring the bucket carefully into a funnel (say), it can take quite a long time to empty, and refilling can be much quicker .....
If all else fails, have a look in the manual - it might just tell you the typical values for the current when charging the battery or watching a DVD. On the other hand, it might not
In the end I think, the only sure way to find out (assuming it is actually possible to do) is to connect an ammeter in series between the charger and the player and measure what current is being drawn under each set of circumstances. Can you run the player from the charger? Without batteries installed
And finally, your leisure battery is probably rated at about 100Ah - which is a simple product of the current and time taken. If your DVD player draws 1A, it will flatten your leisure battery roughly in 100 hours. It works both ways - if you then recharge the leisure battery at 2A, it will take roughly 50 hours to fully recharge. (It's actually quite a lot more complicated than this - of course - but that's roughly how it works.) Likewise with the player's own battery. If you know what the Ah rating is and you know it takes 6 hours to flatten, you can roughly guess what the current is. If you know how long it takes to recharge the battery, you can again work out what the current must be.
I'm not sure I've really been of much help here - you're probably now more confused than you were to start with. Battery technology is a very complex subject, and I've probably been guilty of some quite drastic simplification.
And it was such a simple question
PS: It sounds like you're going to be at home - so why not run a mains lead out of the kitchen window and pretend you're on a hook-up?
