How much do sea levels rise due to thermal expansion?

This is known as the steric component of sea level rise, and it is significant.

As a very rough indication of the size of the effect, if the top 1000 meters of the ocean warmed from 10 to 11 degrees C, sea level would rise by 18 cm. Calculations like this can be made using the equation of state of seawater, which is an empirical thing with lots of terms that gives you the density as a function of temperature, salinity, and pressure. You can read all about it in this article.

We can get the volumetric expansion coefficient $(\frac{0.000214}{C})$ from the link below. I'm going to run through this example for a $1\times1\times 3500 \space \mathrm{m}$ rectangular volume. Like the previous answer, I'll assume a one degree C change in temperature. 3500 m is the average depth of the ocean according to 2 seconds of google.

$V_1 = 3500 \space \mathrm{m}^3$

$\Delta V_2 = V_1\times c_w \times \Delta C$

$\Delta V_2 = 3500 \times 0.000214 \times 1 = 0.749\space\mathrm{m}^3$

So, for a one degree change in temperature, and the average ocean, there is a $0.749\space\mathrm{m}^3$ increase in volume. Our initial 1x1 area makes the sea level rise calculation very easy:

$\frac{0.794\space\mathrm{m}^3} {(1\times1)\space\mathrm{m}^2} = 0.794\space\mathrm{m}$ increase in sea level.

The area ends up cancelling out on both sides, so the answer should be the same for 100x100 or 1000x1000 meter area. The biggest assumption here is that the 'sides' of the ocean are completely vertical. I don't have nearly enough numbers to do the full calculation with shorelines, but this is where the numbers come from.

Coefficient of expansion reference:

The answer is complicated and not a single number. The amount of thermal expansion for a given volume of ocean depends on depth (pressure), salinity, and temperature. That is, a 1C change from 10C to 11C will produce a different amount of expansion than 21C to 22C. What this means is that if you take a fixed quantity of energy (like, say, a gigajoule) and inject it as heat into different parts of the ocean, you will observe different amounts of thermal expansion. Thus, even an "average" warming of 1C across the entire ocean can manifest as a surprisingly wide range of thermal expansions, depending on the particular temperature deltas within each part of the ocean. That is, the final expansion will depend on the distribution of the thermal change.

The best we can do is build models which attempt to make reasonable inferences of the heat distribution and compare them to actual measurements taken by buoys, satellites, and other instruments. This is how the IPCC estimates the thermosteric component of mean sea level rise: Some studies suggest that atmospheric-driven thermosteric expansion can be inferred down to 600m, while others suggest it penetrates further.

The EPA says that the average sea surface temperature (SST is a significant area of study all by itself) has increased on average about 0.07C/decade for the last century: Now, that probably doesn't sound like much change. Barely over half a degree for a whole century! But you need to consider that water has significantly higher density and specific heat than air, so the amount of energy it takes to warm a kg of air is much less than needed to warm a kg of ocean, let alone a m^3.

IPCC estimates conclude that the ocean has absorbed about 93% of the excess energy trapped during the last century or so: What this means is that 1C of atmospheric warming only corresponds to a small fraction of a degree of oceanic warming (hence, the 0.07C/decade).

Even so, thermal expansion is believed to account for at least half of all sea level rise in the last century or so. But to answer your question most directly, NASA documents that from 1970 or so, the ocean has risen about 5 mm/decade, with a corresponding ocean temperature increase of 0.015C/decade (note the different rates cited are due to vastly different timescales). This implies an observed ratio of about 333 mm/1C average ocean temperature increase in the top 700m of ocean. This is about 2x the rate given by @Ben51, and less than half given by @Anthony Herrera.