I want to charge a 12 volt deep cycle battery(wall mart type $40.00). 
Can I hook it up to my car battery and charge it while I am driving or
will this ruin my alternator.  Or would it be better to use a solar
panel. what is the watt capacity for this battery.

Deep cycle batteries should not be charged off a car alternator. 
While you can do it, if you take proper precautions you wouldn't
damage the alternator, it might damage the battery.

No one can tell you the proper way to charge an unnamed battery
without having more info than you put into your question.  The term
watt capacity makes not sense, since wattage is power, while capacity
implies either energy or charge.

I'd suggest the easiest thing to go is do your own battery search, or
rewrite and substantially up your price.  I'm not a researcher, but I
know batteries, and to get useful info to you, it would take a lot
more time than a $2 question would be worth.

Car alternators are generally unregulated, putting out anywhere
between 13 and 20 volts.  Also, if the battery is uncharged to begin
with, the alternator might put out so much current (to maintain 12V at
the posts) that your acid battery might explode.

It is not likely to damage your alternator. The deep cycle battery
could get hot. Check occasonally and disconnect it, if it feels hot
enough you can not hold your hand on it for 20 seconds. More likely
the charge rate will be low, so it will take many hours of driving to
reach a full charge. There is some chance that the regular car battery
will not charge properly with the deep cycle battery connected. If you
make the connections with #20, 22 or 24 wire excessive charging
current is very unlikey as the wire will melt if it becomes excessive.
You need to consider what the wire might touch if it melts. You should
stop charging the deep cycle battery if the voltage (while not
charging) exceeds 13.1 volts. That may however be over charged. The
full charged voltage varies with make and model and goes down near the
end of useful life of the battery. Excessive charging is bad for
batteries and is often difficult to determine. The battery I have is
rated 21 amp hours. it is a sealed battery, not necessarilly for deep
cycle service. A pair of 34 amp hour batteries for a disabled mobility
scooter charge to 14 volts, before the charger shuts down. This may be
over charging.  Neil

New batteries will typically supply 1000 watts for one minute at about
10 volts or 100 watts at 12 volts for perhaps 20 minutes. About half
this many watts near the end of useful life. A battery that weighs 60
pounds may supply double that wattage.  Neil

If it's a 12 Volt deep cycle battery, such as the kind used for
trolling motors, etc., you will probably have decent luck charging it
with a trickle charger that you can find at most auto parts stores.  I
bought one a couple of years ago for less than $10.  The charger plugs
into the wall and has two aligator clips to connect it to the battery.

Be sure to charge the battery in an adequately ventillated place.