You do not specify what type of problems you are experiencing, which makes this a bit of a guessing game. Here are a couple common issues:

Yes, you have a freewheel, not a freehub. Putting a new chain on a worn freewheel often leads to problems, specifically, when the chain is on the smaller cogs on the freewheel it will often fail to seat fully and seem to "skip" particularly when applying a lot of pressure on the pedals. If this is the type of problem you are experiencing, the solution is to replace the freewheel as well, so the new chain will properly seat on unworn cogs. Be sure to get a freewheel with the same number of cogs as your current freewheel to avoid having to replace hub axles, re-space the axle nuts, re-dish the rear wheel, etc. Your picture seems to show a six-cog freewheel, which should be readily available. To minimize problems with chain jamming during front shifts, get a chain designed for seven or eight cogs, rather than nine of more cogs. Chains for six-cog blocks are not advertised as such any more, and a seven or eight cog chain should work fine.