You need to CE mark the PC but this doesn't necessarily mean you have to pay for third party EMC and environmental tests, because the individual components are already tested (or so they say).
You might be able to reason in your technical documentation that placing the components next to each other will not create any EMI scenario that the component manufacturers did not expect. This would include things like wires or metal casing not causing unintended EMI phenomenon by acting as antennas or generating fields etc. Similarly, the PSU needs to be encapsulated so that the customer can't touch the mains voltage even if they open up the computer. If that's not the case, I would imagine that you will have to do a new IP test of the outer PC casing.
The EU directives that you need to address in your EC declaration of conformity should be RED, LVD, EMC and RoHS. Each such directive has a lot of rules and normative standards, but they also have a way to demonstrate compliance without enlisting a test house. Example from the RED directive 2014/53/EU (article 17 §3) where you have 3 different options of demonstating compliance:
(a) internal production control set out in Annex II;
(b) EU-type examination that is followed by the conformity to type based on internal production control set out in Annex III;
(c) conformity based on full quality assurance set out in Annex IV.
If you go with option a) you then need to do the following:
CONFORMITY ASSESSMENT MODULE A
INTERNAL PRODUCTION CONTROL
Internal production control is the conformity assessment procedure whereby the manufacturer fulfils the obligations laid down in points 2, 3 and 4 of this Annex, and ensures and declares on his sole responsibility that the radio equipment concerned satisfies the essential requirements set out in Article 3.
The manufacturer shall establish the technical documentation in accordance with Article 21.
The manufacturer shall take all measures necessary so that the manufacturing process and its monitoring ensure compliance of the manufactured radio equipment with the technical documentation referred to in point 2 of this Annex and with the relevant essential requirements set out in Article 3.
CE marking and EU declaration of conformity
4.1.The manufacturer shall affix the CE marking in accordance with Articles 19 and 20 to each item of radio equipment that satisfies the applicable requirements of this Directive.
4.2. The manufacturer shall draw up a written EU declaration of conformity for each radio equipment type and keep it together with the technical documentation at the disposal of the national authorities for 10 years after the radio equipment has been placed on the market. The EU declaration of conformity shall identify the radio equipment for which it has been drawn up. A copy of the EU declaration of conformity shall be made available to the relevant authorities upon request.
The manufacturer's obligations set out in point 4 may be fulfilled by his authorised representative, on his behalf and under his responsibility, provided that they are specified in the mandate.
That being said, in addition to performing tests, test houses are also very good at helping out with all this EU bureaucracy. You can usually hire them as consultants to tell you what needs to be done, documentation-wise.