Remove lubricant from seat tube and seat post
A thorough way to do it would be to:


Remove the cranks, bottom bracket and seat post.
Make sure there is a decent hole between the seat tube and bottom bracket shell (might depend on the frame).
Ram rags down the seat tube with a broom handle and extract them via the bottom bracket shell.
Once the bulk of the grease is removed (or pushed to the bottom of the seat tube) soak a rag in degreaser (WD40, white spirit, citrus degreaser etc) and ram it through to remove the last traces.


This would be a bit of a pain though and you might not have the tools or knowledge to remove and refit a BB.

You could instead try tying a strong cord to the rag so that you can ram it down the seat tube then pull it back out. If the cord breaks or slips though you'll be back to removing the BB or you will just have to leave the rag there.

You should also ask why the seat post is slipping, as they shouldn't do so even when greased. Actually, it is normally recommended that you grease seatposts lightly so that they don't seize in place. Could there be something wrong with the seat clamp? Alternatively, seat posts tend to slip when they are slightly too small for the seat tube. It seems annoyingly common for manufacturers not to spec the seat tube diameter properly but it could be that the previous owner replaced the seat post with one that is too narrow?

If you are dealing with a screw-type seat post clamp (as opposed to a quick release) and you are doing it up as tight as you can in an effort to stop it slipping then I would strongly recommend that you carry a spare one with you. I have broken a seat post clamp doing the same thing and it was quite a miserable cycle with no saddle to the nearest bicycle shop 10 miles away.

If you have a quick release seat post clamp my gut feeling is that you could improve your clamping force by moving to a screw-type one. Get one with a large diameter screw if possible but make sure that screw is made of steel! Some use aluminium for the screw which is garbage. Also preferably get one where the screw threads into a steel insert rather than into the aluminium body of the clamp.

Edit: As the commenters have said, carbon frames should not be greased. They use an anti-slip or 'carbon assembly' paste. This paste might help you even if your frame is not carbon; I've never managed to get a straight answer on whether it works for metal-on-metal contact.

Also the double seat post clamp linked by Carel Looks brilliant. I've not seen such a thing before.