The use of the relativistic heavy ion collision experiment has extended our insights into the diverse possibilities available to a truly strongly-interacting system. The main goal of this experiment is to describe the properties of the different phases of quantum chromodynamics (QCD) and to chart the QCD phase diagram on the T-mu plane. For the phase diagram, apart from the general phase boundary lines, some specific characteristics such as the possible critical endpoint (CEP), associated coexistence region, and strongly-coupling quark-gluon plasma (sQGP) have to be identified. Here, the CEP separates the first-order phase transition from the second-order transition (or crossover) when the case beyond the chiral limit is considered. However, convincing signals have not yet been obtained using the relativistic heavy ion collider (RHIC) experiment. Theoretically, strong interaction systems hold significant features: asymptotic freedom in the ultraviolet region, dynamical chiral symmetry breaking, and confinement in the infrared region. Such features can be uniformly displayed in the phase structure of the matter in the temperature T and chemical potential planes. Consequently, several investigations have been experimentally and theoretically performed. However, the strong coupling feature in the low-energy region prevents the use of perturbative calculation methods, which creates the need for the development of nonperturbative approaches. Additionally, lattice QCD simulations have been widely implemented; however, the "sign problem" delays the progress in the large chemical potential region. Therefore, the Dyson-Schwinger equation (DSE) equation method and functional renormalization group approach, which inherently include both dynamical chiral symmetry breaking (DCSB) and confinement, play an important role. The QCD DSE approach is a method based on the continuum quantum field theory. The new criteria were proposed based on the DSE and studied using the deconfinement and Chiral symmetry restoration phase transition of QCD. Currently, functional methods can be used to provide a reliable estimation of the CEP location. First, reliability is achieved using a thorough investigation of the truncation of the DSE, state of the art truncation is then performed causing a converging result between the different methods, and the predication of the lattice simulation at low chemical potential is confirmed. The results show a fast convergence of the truncation owing to the infrared fixed point of the QCD coupling, which allows the capturing of the QCD running behavior using a finite set of two- and three-point Green functions. The estimated location of the CEP based on the current computation is μ_B at 600~650 MeV and T at 100~110 MeV. The existing functional QCD methods are non-perturbative continuum methods that are capable of simultaneously describing both the DCSB and confinement. Although they are limited by the truncations, the use of functional QCD approaches has resulted in progress in the study of the QCD phase structure and thermal properties, where a complete phase diagram and related thermal properties have been obtained in a large chemical potential range, which can provide a reference for the exploration of the QCD features. Most of the theoretical studies using effective models or certain truncations have observed the existence of the CEP; however, the determination of its location is still a work in progress because it varies based on the computation. Moreover, searching for QCD phase transition signals, particularly the CEP, is the main goals of current and future experimental programs on the relativistic heavy ion collider.