I'm an Angevin specialist and can answer this from that perspective. However, I would welcome additions from someone with access to HRE primary sources and historiography, as my analysis draws exclusively from an Angevin point of view.

Unfortunately, we don't have specific expenditure records of how the ransom money was going to be spent. Medieval rulers tended to think that financial records were not particularly glamorous, and of little interest to their descendants. Financial records were therefore kept on degradable materials and care was not taken to their preservation.

We know that the price for Richard I's ransom was 100,00 marks, half of which would go to Duke Leopold of Austria, and half of which would be kept by the Emperor Henry VI. Additionally, they wanted troops and arms supplied by Richard, and in fact, led by Richard, in Henry VI's conquest of Sicily. Leopold also wanted the release of his Comneni relations that Richard had driven out of Cyprus, which was granted, and a betrothal between one of his sons and Eleanor of Brittany, which was also granted. In order to ensure that this agreement would be honoured, Henry VI and Leopold demanded 200 hostages, of which Leopold kept 50.

The payment of 100,000 marks was eventually paid and Richard was released. Isaac Comnenus and his daughter had also been released previously. Arrangements were made for Eleanor of Brittany to be sent to Austria. However, before this could be carried out, Leopold died from a botched amputation of a gangrenous foot, so the marriage was not honoured. Furthermore, Leopold had also been ordered by Pope Celestine III to pay back the ransom money, though by his time of death it had not been paid, and I did not find record that it had ever been paid.

As for Henry VI, John Gillingham assumes that he spent the majority of his funds campaigning against rebels in the Rhineland, and subjugating parts of Italy. He also completed his conquest of Sicily. However, as far as I can tell, Richard never supplied the additional troops that had been requested. It probably would have been pushing his luck for the Emperor to have pressed that agreement any further considering the dubious legality of the ransom under Ecclesiastical law.

Unfortunately, there is not a very concise record of how the ransom was spent. John Gillingham writes that by 1194, Leopold of Austria had spent 26,000 marks, but does not comment on how these funds were spent.

The best source for Richard's life, and his captivity and ransom, is John Gillingham's biography, which is widely available in print. Biographies of his mother, Eleanor of Aquitaine, by Ralph Turner and D.D.R. Owen corroborate Gillingham's summary.

I would love it if someone who knows this period of history in the HRE could provide some insight from that perspective! But I hope this helps in the meantime!