What percentage of modern day Turks are of Bulgarian/Circassian/Greek/Serbian descent?

For the record, I am neither a Geneticist, nor a Genetic Anthropologist; therefore, my answer, is based upon the historical observation of Turkish demography throughout the Modern era.

The Turks originally came from Mongolia. They, like the Huns and Magyars centuries before, were Mongolian warriors who traveled through the continent of Asia-(probably via the famed Silk Route) en route to the European continent. Although the Mongol-Turks were fierce and formidable warriors, they were relatively small in number when compared with the much larger indigenous populations of the various countries and lands they conquered.

Although the early generation of Mongol-Turkish warriors were outnumbered, they did intermarry with women from the conquered indigenous populations. However, the real significant demographic and cultural change, was the process of Ottomanization; that is to say, a process of forced religious and linguistic conversions, especially within the centuries old Christian land of Asia Minor-(present-day Turkey). The majority of Greek and Armenian Christians living in Ottoman ruled Asia Minor, became Muslim and spoke Turkish, thereby losing their centuries old Christian, Hellenic and Armenian cultural identities...... specifically, their religion and languages. (However, there were smaller percentages of Asia Minor Greek and Armenian Christians who were able to survive the process of Ottomanization through the centuries).

The most famous example of Ottomainzation were the Janisarries who were the most ruthlessly and ferociously trained Elite Fighting Corps within the Ottoman military-(until 1826). The vast majority of the Janisarries were originally Eastern rite Christians from the Balkans. Romanian, Serbian, Bulgarian and especially Greek men, were essentially drafted into the elite Ottoman Fighting Corps and in doing so, were thoroughly Ottomanized. (The Romanian Prince Vlad Dracul.....a.k.a. "Dracula's" brother......was a Janissary).

Currently, there are approximately 70 million ethnic Turks residing in Turkey. It is a fairly large country in landmass and has always been at the geographical, imperial and cultural crossroads. Many populations have resided in Turkey through the millennia dating back to the Ancient Hittites. The historical populations of Turkey have included, Greeks, Armenians, Romans, Persians, Kurds, Georgians, Hittites, Goths, Slavs, Jews, Syrians and of course, Mongol-Turks. Of the populations listed, the Mongol-Turks were the historical latecomers, as well as one of the smallest in population percentage. Though of the populations listed, the Kurdish, Armenian and especially, Greek populations, were the oldest and largest in population percentage. (And in the case of the Greeks and Armenians, they were the most vulnerable for imperial conquest and conversion, though the other above mentioned populations certainly experienced the process of Ottomanization over the centuries as well).

Today's Turkish population reflects much of the demographic diversity of its 4000 plus year history whereby few of the original Mongol-Turks have survived. The majority, indeed the vast majority of present-day Turks, are of diverse ethno-racial backgrounds, with a sizable percentage of peoples who were originally of distant Armenian and Greek Christian ancestry.

I am not sure if it is possible to define a percentage of the population as descendants, as it is quite challenging to define a descendant of one ethnic group scientifically, when you also need to define that other ethnic group scientifically.

There are several studies on DNA distribution of populations on many countries. One of the serious studies on Turkey is "Excavating Y-chromosome haplotype strata in Anatolia" by Cinnioglu et al., Stanford University.

It shows the following results as the Y-DNA haplogroup distribution of the people of Turkey:

E1b1b1 = 10.7% (common in the Mediterranean region)

G = 10.9% (common in the Caucasus, also found in the Middle East)

I = 5.3% (common in Central Europe, the Western Caucasus, and the Balkans)

J1 = 9% (common in Arabia and Daghestan)

J2 = 24% (common in Western Asia and Southeastern Europe and also found in Central and South Asia)

K = 4.5% (common in Asia and the Caucasus)

L = 4.2% (common in India and Khorasan)

N = 3.8% (common in Eastern Europe and North Asia, including Siberia [e.g. Turkic-speaking Yakuts], the Altai Mountains region, and the Ural Mountains region)

Q = 1.9% (common in North Asia including Northern Altaic peoples)

R1a = 6.9% (common in Central Asia, the Caucasus, Eastern Europe, and among Indo-Aryans)

R1b = 14.7% (common in Western Europe)

T = 2.5% (common in the Mediterranean, South Asia, and Northeastern Africa)

You need to pay attention that Y-DNA is only showing the male ancestors, and not female - though I am not sure how much difference would it make.

Also important to note is that above is the Y-DNA distribution in the population, and the population is not formed of people under one of these categories. Everyone has multiple of these categories, as all people carry genes of thousands of ancestors.

So in many regards, Turkey has a very diverse ethnic profile, that is very much mixed. You may also compare the Y-DNA profiles of other countries in Europe in the map of the following study, where you will see the profile is more or less very similar to neighboring countries such as Greece.