The first and easiest thing to do is measure the conductivity, which can help you get a good estimate of the inorganic salts present. To determine the total organic carbon (TOC), typically an analyzer like one of Shimadzu's is used, with a high temperature combustion method. See the attached paper. That is easy if you have the analyzer; not as easy if you don't!
Well, I think only need to check for the organic part since seawater definitely has inorganic compounds as salts. For organic part, a quick method is COD determination which takes abour 2 hrs.
What about DGT – diffusive gradient in thin films; a passive sampling technique. It can be used for simultaneous determination of inorganic/organic fractions in water.
If you just take 100 mL sample and evaporate it at 103oC and take out all water. The residue is called total solids consists of organic and inorganic. The organic portion of the TS is called Volatile solids and inorganic (ash) portion is called fixed solids. The volatile solids analysis is carried out from the evaporated residue (TS) to be kept in 550 oC for 6 to 12 hours and volatilise all carbon , the loss on ignition is organic portion and residue (ash) is an inorganic portion. You can simply weigh the Ash and the TS-ash= organic portion, (This is available in the book STANDARD METHODS FOR THE EXAMINATION OF WATER AND WASTE WATER 20TH EDITION BY AMERICAN PUBLIC HEALTH ASSOCIATION.