This paper presents the work carried out by the ASASVIcomtech team, made up of researchers from Vicomtech and University of Granada, for the ASVspoof5 Challenge. The team has participated in both Track 1 (speech deepfake detection) and Track 2 (spoofing-aware speaker verification). This work started with an analysis of the challenge available data, which was regarded as an essential step to avoid later potential biases of the trained models, and whose main conclusions are presented here. With respect to the proposed approaches, a closed-condition system employing a deep complex convolutional recurrent architecture was developed for Track 1, although, unfortunately, no noteworthy results were achieved. On the other hand, different possibilities of open-condition systems, based on leveraging self-supervised models, augmented training data from previous challenges, and novel vocoders, were explored for both tracks, finally achieving very competitive results with an ensemble system.