Did classical composers such as Tchaikovsky and Beethoven have other occupations or did they just come from a wealthy background.

I can answer your question for Beethoven with the sources I currently have on hand. [Edit: I've added Tchaikovsky.]

Composers in the 19th century were, as they had been in previous centuries and still were through our 1995 cutoff date, usually practicing musicians of one kind or another. They were able to make a living as musicians, sometimes even a relatively lucrative one. In the 18th century, continuing into the 19th, many of the composers whose names we remember today worked as members of a court musical establishment or for a well-endowed church. The primary duty of such work might be instrumental playing, singing, or administrative, as in the case of the Kapellmeister (chapel-master), the highest-ranking position. Composition of new works was generally part of the expected duties only for the Kapellmeister. Other sources of income for composers included sale of works to publishers, private teaching, and subscription concerts, in which the composer acted as entrepreneur by selling tickets, renting a venue, and hiring musicians, allowing him to keep any profits.

Beethoven was one of the earlier composers to make a living without the benefit of a court position as his main income source; he could almost certainly have been hired as a Kapellmeister, but preferred what he considered to be the greater artistic freedom of a freelance practitioner (it should also be noted that for various reasons, court budgets in German-speaking areas were frequently being cut at this time). His grandfather had been a Kapellmeister in Cologne, and his father was a singer for the archbishop's court in Bonn. In Beethoven's case, this quotation from a 2012 article sums the situation up fairly well:

Beethoven made much of his money in five ways:

playing the piano and conducting in public and private concerts;

composing on commission;

composing music that was sold to publishers;

donations from kings and the aristocracy; and,

private pupils[.] (Staffan Albinsson, "Early Music Copyrights: Did They Matter for Beethoven and Schumann?," International Review of the Aesthetics and Sociology of Music 43, no. 2 (December 2012): 280.

Most of the concerts would have been subscription concerts, but before he became hard of hearing, Beethoven was a sought-after concert pianist who would have been engaged by other concert promoters also. Beethoven was far from alone in receiving "donations" as a type of patronage (of course, court positions were themselves patronage as well as employment, since the tasks involved were non-essential). Beethoven's were unusually lucrative, however. For example, after 1810, Prince Kinsky (two generations of Prince Kinskies, actually) paid him, at least once in most years, amounts that ranged from 90 to 3000 florins. (Source: Donald W. MacArdle, "Four Unfamiliar Beethoven Documents," Music & Letters 36, no. 4 (October 1955))

How much was a florin? The worth of money naturally depends on how much people get paid and how much things cost.

Salaries

(Source unless otherwise noted: Alice M. Hanson, "Incomes and Outgoings in the Vienna of Beethoven and Schubert," Music & Letters 64, no. 3/4 (July-October 1983))

The Kapellmeister in Vienna was paid around 2000 florins per year between 1815 and 1837. A performer in the state theatre orchestra averaged about 250 florins per year. The violin instructor at the state conservatory made 700 florins per year. Musicians frequently held at least two such positions, perhaps one as a performer and one as a teacher.

For the rights to several works from a top-selling composer like Beethoven, a publisher might pay what were therefore quite high sums; in 1810, for 9 works, Beethoven's publisher paid him 200 ducats (MacArdle). A ducat was worth 4.5 florins in the 1820s, so even in 1810 that was probably between 700-800 florins.

For a comparison to other professions, school teachers made from 400 to 900 florins annually, soldiers 30 to 50, most officers about 230 to 1800, and male factory workers 160 to 320 (14-hour day). Beethoven himself paid two servants a total of 249 florins in 1823.

Prices

For an idea of what standard of living these salaries might buy, we can look at housing prices, which varied considerably but are nevertheless instructive. In 1810, one room rented for 100 florins per year in a suburb of Vienna. In 1811, one could rent a 6-room apartment in the outer suburbs for 150-200 florins annually. In 1824, a 3-room apartment in the suburb of Hietzing ran 260. Suburbs were less expensive than the city itself. In 1827, depending on the location, one room might run from 52 florins in Liechtenthal to 441 in the inner city (which was the most desirable, remember).

In 1817, a court laborer wrote that he needed about 1,000 florins per year to meet his expenses, which were lodgings in a far suburb, food, and fuel.

A practicing musician, thus, appears in Beethoven's time and place to have earned a living somewhere between that of a laborer and a teacher or military officer, with a successful composer on the high end of that or higher. It was a sufficient occupation on its own, and didn't require one to have other lines of work or independent wealth.

Tchaikovsky was born upper-middle to upper class. His father was a mining engineer. He was educated at the St. Petersburg School of Jurisprudence, and worked at the Ministry of Justice there as a young man. After he finished his musical studies, he taught at the Moscow Conservatory from 1866 until 1878, while also earning some money from publishing and arranging. He most likely would have continued in this fashion, but in 1877, a wealthy woman named Nadezhda von Meck decided to become Tchaikovsky's patroness; she provided him with a regular stipend thereafter. Beginning in the late 1880s, he also began to appear as a conductor. (Source: any biography of Tchaikovsky, but in this case, R. J. Wiley, "Tchaikovsky, Piotr Il'yich," Grove Music Online [subscription required].)