The short version would be, The President and High Representative are chosen by the Nation States collectively and approved individually by MEPs. Each nation not represented by one of these posts forwards a candidate individually for the Commission, who are then approved collectively by MEPs. Finally the President selects their own Vice-Presidents which must include the High Representative.
The linked Wikipedia Article seems to answer this question quite fully.
First the President is nominated by the Heads of State selecting a candidate through mutual negotiation. That candidate is then put to a vote of the European Parliament (MEPs).
The President of the Commission is first proposed by the European Council taking into account the latest Parliamentary elections; that candidate can then be elected by the European Parliament or not. If not, the European Council shall propose another candidate within one month.
Following the link to High Representative on the first page gives much the same process as for the President.
The High Representative is appointed by the European Council acting by qualified majority. However to take up their role in the Commission, in particular as a vice-president, the High Representative has to appear before Parliament for questioning and then be subject to Parliament's vote of approval on the proposed Commission.
Each member state then nominates a Commissioner, the whole of which collectively is approved by a vote of the European Parliament.
Following the election of the President, and the appointment of the High Representative by the European Council, each Commissioner is nominated by their member state (except for those states who provided the President and High Representative) in consultation with the Commission President...
The President's team is then subject to hearings at the European Parliament which will question them and then vote on their suitability as a whole...
As Parliament cannot vote against individual Commissioners there is usually a compromise whereby the worst candidates are removed but minor objections are put aside so the Commission can take office. Once the team is approved by parliament, it is formally put into office by the European Council (TEU Article 17:7).
Finally the President appoints the Vice-Presidents under their own authority.
Following their appointment, the President appoints a number of Vice-Presidents (the High Representative is mandated to be one of them) from among the commissioners.