What is the relationship between the trade deficit and foreign debt?

These two concepts are not directly related. In order to understand what trade deficit is, you need to look at the balance of payments (BOP) equation:

Current Account + Capital Account + Financial Account (+ Balancing Item) = 0

What we usually mean by trade deficit is a deficit in the current account, when a country imports more goods and services than it exports. 

From the above equation it is observable that a deficit in the current account will be balanced by a surplus in the capital account, which means a lot of the U.S. dollars used to pay for net imports are invested back in U.S. capital assets while a smaller portion is used to pay for U.S. exports. This is because foreigners can't readily spend U.S. dollars in their domestic markets.

So far, you will see that this has nothing to do with foreign debt, which is accumulated when foreign investors want to buy U.S. Treasury debt and U.S. corporate bonds - because the U.S. dollar is still the world's most prevalent reserve currency and it is much safer for a rich third-world tycoon to hold U.S. debt than that from his native country.  

Foreign debt in a nutshell, is just a portion of a country's total debt, as part of national debt is financed domestically and the other part externally. High foreign debt can be a problem as it is not taxable by the domestic government, and interest and debt payments do not flow back to the domestic economy, so it is crucial that foreign debt be at a reasonable level of GDP. It is also easier for countries to roll over internal debt than external debt (in theory, we see the U.S. rolling over much foreign debt all the time). 

It is also important to keep the current account deficit in check as persistent deficits reduce GDP. However, trade deficits  do not grow foreign debt. Although, it can be reasoned that both trade deficits and high foreign debt levels are caused by a country consuming beyond its income and failing to make productive investments.