I'm just a shade tree bicycle mechanic, but it sounds like you just need to tighten the headset adjustment. Assuming this is an older threaded headset there are two big nuts on top. The lower one may have flats for a wrench or be knurled for finger tightening. Turn that one until it contacts the bearings. You don't want it tight, just enough contact to get out any play. You then have to hold that one still and tighten the locknut against it. This takes some large wrenches, 32-36 mm. If you are not clear on this stop by your LBS, and let them do it. Some places will do something like this free because it only takes about 5-10 seconds if you have the tools and know what to do. The grease is just a symptom. If you do take it to your LBS you might want to go ahead and service the headset (clean, replace bearings, repack), especially if the bike is old. They probably don't charge much. This is not a bad job to get your feet wet in bike mechanics, but if you decide to, please get a book for a complete description. To avoid potential damage, if you don't have appropriate wrenches you can actually just hand tighten the nuts as described to lessen the amount of play immediately.

If your headset is threadless the procedure it different.