When it comes to "best tow vehicle" such as your title, I was thinking for what size load and how many miles to travel each year.
Several have converted 4-8 year old class 8 trucks from 3 axle to 2 axle and have a great truck that can tow just about anything with a 400 HP diesel and 9 to 13 speed automatic transmission. These cost about $34,000 once you sell the extra rear axle and spare tires. While these trucks already have 350,000 miles on them, many will run 850,000 miles before taken out of line service when they are only about 8 years old. Yes driven more than 100,000 miles a year for 8-10 years is not uncommon anymore.
Yet with a $5,000 budget, you are not looking for the "Best" but rather the most economical. I don't know what to say, because someone willing to sell a diesel truck for $5,000 means they already know it is a money pit, they are getting rid of it for that reason, and you will probably pay $50 to have it inspected just to find out that the transmission might be slipping or the engine is about toast.
You might be better off paying $4,000 down and payments of say $150 a month for 2-3 years to buy a slightly better truck. You might be buying a truck that will have less breakdowns. This might save you more than the $150 monthly payment.
I agree that looking for a 7.3L diesel truck would be a great idea. Also consider a 7.3L equipped transit van, like a airport shuttle van. Complete with a wheelchair lift that might be sold for say $1,250 or so, you can sometimes buy a used airport shuttle van with about 200,000 miles on it for about $7,500 - $8,000. Getting a diesel, you would want to look for a E-450, with it's heavier 14,050 pound GVWR. The lighter duty E-350 would only have a 12,000 GVWR, and that heavy 800 pound diesel on the front axle means it might not have much cargo capacity for towing a trailer.
Doing a search for E-350 and Diesel fuel will quickly find some on TruckTrader.com.
I also agree that the older V10 Fords and 8.1L GMC's and even the Dodge V10's had great pulling power, just poor miles per gallon. This leaves their sales price fairly low cost, while driving costs will be higher per mile. This will work well if you are doing less than 4,000 miles a year.
Fred.