First you have to define what you mean by dissolved inorganic carbon (DIC). This is often understood as just the sum of carbonate, hydrogen carbonate ("bicarbonate") and CO2 or carbonic acid. In other cases less common species are included, e.g. cyanide, cyanate, thiocyanate, carbamate, thiocarbamate etc. If the latter is the case, most of these compounds can be converted to carbonate by oxidation and/or alkaline hydrolysis.
Then you have to figure out qualitatively what else you have in your solution. It's very rare for an analytical method to be free of potential interferences, and you need to be aware of possible problems.
The next step is usually (a) to make the solution basic enough to convert all DIC to carbonate, then to (b) precipitate it as insoluble carbonate with a soluble alkaline-earth salt such as CaCl2 or Sr(NO3)2; (c) separate by filtration; (d) acidify to decompose the precipitate and collect evolved CO2; and (e) quantify the amount of gas (which may or may not involve a titration method). There are many variations, but this is a multi-step procedure.
However, you can also titrate the pH-modified sample directly after step (a) above, using a soluble alkaline-earth (Ca/Sr/Ba) salt to precipitate the respective carbonate. The trick is to have the right end-point determination method available to identify the point at which precipitation is complete. (You are surely familiar with precipitation titrations of chloride by silver nitrate; this is just another special case.) There are several options, including:
thermometric (good for relatively high concentrations);
conductometric;
potentiometric with an ion-selective electrode; or
turbidimetric, with a colorimeter connected to a probe immersed in the titrated solution.
In all these cases, the end point is revealed by a sharp break in the curve defined by the relevant physical property - respectively, solution temperature; conductance; potential; or absorbance. If you use a titration system capable of calculating derivatives, the end point is easily defined by the second derivative of the titration curve. (If not, you can often "eye-ball" it very accurately.)
Here is a partial example of a carbonate determination I performed a while ago; due to confidentiality issues I can't give full details about the sample solution.
Below is the example titration I mentioned in the previous answer.
Note that because of the low CO2 concentration (~35 ppm) a standard addition had to be used, because without the extra carbonate the precipitation was slow. Also note the use of aqueous alcoholic titrant solution - that was done in order to decrease further the carbonate salt solubility and sharpen the end-point.