The first stage of the financing of the Louisian Purchase is stipulated in the treaty itself–France received $11.25 million in American bonds (called “stock” in the treaty, but we would call it bonds today), and $3.75 million in assumption of claims by US citizens against the French government.
The agreement on citizen claims was ridiculous; total claims amounted to far more than $3.75 million, and who would decide which were honored by the American government and which remained pending against France? This was a source of irritation for decades afterward. Nevertheless from an American perspective this did not require special financing; the claims were paid by the Treasury out of ordinary revenue as they were (gradually) presented.
The bonds, as per the treaty, paid 6% interest due semi-annually with the principle being redeemed after 15 years. Note that wikipedia is wrong in asserting an American down payment.
Napoleon needed cash right away, and would not have agreed to take the bonds without assurance from Barings and Hope that he could sell some of them immediately for gold. He got an advance in July 1803 and sold the bonds, after delivery, beginning in 1804; details here.
Yes, the British government objected; Prime Minister Henry Addington wrote this letter to Barings. Barings ignored him, and I am not aware that they faced any consequences.
The Louisiana Purchase raised the US national debt from $70 million to $81 million, at a time when government revenue ran about $10 million per year. However this was quite small in relation to a GDP of about $500 million, in a nation with a rapidly growing population (which the purchase itself helped to stimulate), so economically this was not a huge imposition.