In this thesis project, we study the field of high-dispersion spectroscopy and methods for extracting the spectrum of circumstellar objects such as exoplanets from the combined signal of a stellar system. One of the only techniques for detecting absorption lines in exoplanetary atmospheres is to directly image a planet and record the reflected light. However, exoplanets are incredibly faint compared to the parent star and are often completely obscured in any images of the system. We utilize techniques such as high-dispersion spectroscopy (HDS) and high contrast imaging (HCI) in order to capture the planetary signal and develop methods for reducing only the stellar light while leaving the planet relatively untouched.   We investigate a method for removing the scattered starlight by utilizing the separate spectra of the star and the planet, where the signal from the objects will be spread out according to a point spread function (PSF) and laid on top of each other. By empirically determining the shape of the stellar PSF, reference profiles can be created for each wavelength and subtracted from the entire signal, revealing the planetary spectrum. To achieve this, we have constructed a spectrum extraction algorithm, written in Python 3.6, for use on the spectra of directly imaged exoplanetary systems. Additionally, we discuss many of the problems which may arise when reducing cross-dispersed echelle spectra and attempt to solve them with the algorithm.   To assess our algorithm, we utilize spectral images of the system  Pictoris, taken with the high-dispersion spectrograph CRIRES, and three model exoplanetary systems of varying brightness. When extracting the spectrum of the planets, we find that the method employed for constructing the reference stellar PSFs is partially flawed and leaves a substantial amount of residual stellar light in the reduced images. This leads to difficulties with identifying any spectral absorption lines and an alternative method is likely necessary. Nonetheless, the algorithm is found to successfully extract the spectrum and identify spectral lines of an exoplanetary atmosphere if the planet is sufficiently bright, although only for theoretically unrealistic luminosities. We expect that our algorithm can be improved upon with more well-researched methods for reducing the starlight and by using data recorded with spectrographs of even higher dispersive capabilities, such as CRIRES+, METIS, or HIRES.