Did the citizens of the Warsaw pact view their government as client states? If so, how did the Russians of the Soviet Union view their client states?
Summary: In Warsaw Pact Czechoslovakia the average citizen thought his country was subordinate to the local "Communist Party of Czechoslovakia" and that mattered for his daily concerns. He knew that by extension the country was subordinate to its supervisor Soviet Union but that didn't affect his daily routine that much, apart from forced participation in praising "brotherly cooperation".
After the Prague Spring events of 1968 and the subsequent Warsaw Pact invasion Czechoslovaks learned the hard way that it is primarily Russians who is in charge. These events changed the attitude of many Czechoslovaks and left a bitter aftertaste that is still alive for many Czechs and Slovaks to this day.
Long decades before Czechoslovakia adopted socialism in 1948, back in the 19th century when Czechs and Slovaks were still subjects of the Austrian Empire and in theory they would possibly look up to Russia with hope, various Czech and Slovak journalists/intellectuals already published warnings against panslavism with Russia as a leader.
I'll just cite one of them, the journalist Karel Havlíček Borovský, who visited various Slavic countries including Russia. He wrote the essay "Čech a Slovan" ("Czech and Slav") which was published in 1846 in the newspaper "Pražské noviny".
Essay fragments in Czech:
"Seznal jsem Polsko, — nezalíbilo se mi, s nenávistí a s pohrdáním opustil jsem sarmatské kraje, a po novém roce v nejtužších mrazech klusal jsem na kybitce do Moskvy, zahříván nejvíce horkostí své srdečné všeslovanské vzájemnosti. Ruské mrazy a ostatní ruské věci uhasily ve mně poslední jiskru všeslovanské lásky; kosmopolitismu jsem bez toho nikdy neměl ani za mák v těle, a tak jsem se vrátil do Prahy Čechem, pouhým neústupným Čechem, ba s jakousi tajnou kyselostí proti jménu Slovana, které mi seznavšímu dostatečně Rus i Polsku jakousi ironií zavánělo."
"A zrovna tak jako Španělé, Portugézi, Francouzové, Vlaší dohromady jsou Románští národové; a Němci, Dáni, Švédi, Norvézi, Hollanďané Germánskými národy slovou, zrovna tak jsme my, Poláci, Iliři, Rusové atd. Slovanskými národy, a jméno Slovan jest a zůstati má vždy jen jménem zeměpisným a vědeckým, nikoliv však jménem srdečným, jménem sympatie, s jakou každý národ jméno své vyslovuje."
Translation:
"I've gotten to know Poland - didn't like it. With hatred and disdain I left the lands of Sarmatia and after a New Year I trotted to Moscow on a kibitka (carriage) through toughest frost, warmed mainly by my feeling of panslavic intercommunity. Russian frost and other things Russian extinguished the last spark of panslavic love in me; I was never a cosmopolite in the first place and so I returned to Prague a Czech, a mere stubborn Czech, even with a taste of an underlying sourness towards the name Slav which smelled of irony to me after getting to know Russia and Poland well enough."
"Like Spaniards, Portuguese, French and Vallachians make together a Roman group of nations; like Germans, Danes, Swedes, Norwegians, Dutchmen are called German nations; so are we, Poles, Illyrians, Russians, etc. Slavic nations, and the name Slavic is and shall remain a name in geography and science only, but not a name cordial, a name of sympathy with which every nation pronounces its name."
You can understand from the above, that Czechoslovaks thought of Russians to be a completely different nation despite the same language group. They already had large experience with them from WWI, since eastern front Czechoslovak legion was formed in Russia and numerous Czechoslovaks learned about Russian ways of life firsthand before returning home after the war. The same applies for WWII.
We can therefore say that after the coup of 1948 when Czechoslovakia turned socialist, it wasn't for panslavism reasons of feeling so similar.
The Warsaw pact was signed in 1955. With Stalin already dead (+1953) and the Czechoslovak president Klement Gottwald also dead (+1953), the toughest era of political murders was over but the grip of the Communist Party of Czechoslovakia stayed strong for another 10 years. People back then were very well aware of the power the ruling Communist Party of Czechoslovakia had over them. They were very well aware that Soviet Union and its Communist Party is the decisive power in "The Socialism Camp" but a power somewhat distant. People gradually came to believe that while they stay on the side of Soviet Union, they can pretty much rule themselves as they see fit. This led to a more relaxed atmosphere in Czechoslovak society in mid 60-ties with loosened censorship and more or less cryptic criticism of Soviets in arts and movies.
The following years up until 1968 saw the Communist Party of Czechoslovakia itself making effort to introduce democratic reforms that would create "socialism with a human face". Their efforts were stopped by the Soviets who decided upon a military intervention where some other Warsaw Pact countries participated (and interestingly some refused).
In any case, after the intervention the Soviet Army stationed permanent garrison in Czechoslovakia and people often changed their attitude toward Soviets even if they were fervent communists before. The following years up untill the Velvet Revolution (1989) were called "normalisation". The era favoured spineless careerists whos only expertise was often just the membership in The Party.
There are some quality Czech movies that depict the socialism atmosphere and/or the relations with Soviet Union really well.
For tough 50-ties, try "Ve stínu".
For normalisation 80-ties, try "Pouta".