We investigate the spectral and transport properties of a double quantum dot laterally attached to a topological superconducting nanowire, hosting the Majorana zero-energy modes. Specifically, we consider a geometry, in which the outer quantum dot is embedded between the external normal and superconducting leads, forming a circuit. First, we derive analytical expressions for the bound states in the case of an uncorrelated system and discuss their signatures in the tunneling spectroscopy. Then, we explore the case of strongly correlated quantum dots by performing the numerical renormalization group calculations, focusing on the interplay and relationship between the leaking Majorana mode and the Kondo states on both quantum dots. Finally, we discuss feasible means to experimentally probe the in-gap quasiparticles by using the Andreev spectroscopy based on the particle-to-hole scattering mechanism.