Just weight carefully a precise volume of solution (for example 1 mL). Measure it ten times and calculate the average. Be sure that you control the temperature since the density is also dependent on the temperature.
I agree with Daniel's recommendations and just add the following comments to perhaps further help you with your pre-planning of the experiment.
Much depends on how many decimal places you require your measurement to reliably report. You can check with scientific vendors such as "Fisher Scientific", for instance, and search for "pycnometers". Some of these can be simply calibrated at a given temperature (use a few different standard diluents to do this, or sucrose/water mixtures) and then applied to unknowns. The simplest pyncometers are glass with a capillary, but some also include a thermometer. These pycnometers would most typically be used with a four place analytical balance.
As Daniel has pointed out, you could use micropipettes or other volumetric delivery (To Contain --TC, not To Deliver TD) devices with a four place balance to approximate your densities.
If you are seeking to track the density through the critical micelle concentration, or other more exacting work, your temperature control requirements along with your pycnometer requirements (or u-tube vibrating densitometer) will become much more stringent.
I agree with Daniel and MIke. If you have access to a U-tube density meter available, I would recommend using it, since they usually have built in temperature control over a reasonable range of temperatures, and the precision will match and exceed what you can get volumetrically/gravimetrically. We use an Anton-Paar DMA-4500M (Graz, Austria) u-tube instrument.
Density/Sp. Gravity measurement for surfactant solutions - sp. gr. bottle is good enough. As a precaution, check for entrapped bubbles if there is any.