I'm a bit opaque on when that actually is. If you mean circa 1965, it is probably too late or at least too hard. The Soviet's best chance for a puppet Congolese regime needs changes from '64 at the latest. The Soviets need to convince their client, the Simbas (People's Republic of Congo if you prefer) to not take hostages and so avoid any Belgian-American intervention (Dragon Rouge et al). If they displace undesirables without harming them (big, big if, I know), the western powers may be disinclined toward direct involvement. If they can subsequently maintain a relatively stable regime in Stanleyville beyond the end of '64, a play for the rest of Congo becomes a lot easier down the line.
It may be fun to imagine how the Soviet Union goes about preventing western intervention. The Belgians would seem to be the weak link (no offence to Belgium but I'm comparing them to 60s US military might). Perhaps by shadowing Belgian trooping flights so they get the message or perhaps more fancifully, a shoot-down event. Either by carefully positioned cruiser over the Med or somewhere across the vast tracts of empty desert. As I said, perhaps a bit fanciful. I think a would-be PRCongo might rely more on western apathy and bluff than on the Soviets truly going to bat for them but stranger things have happened.
If successful in maintaining a foothold, they would still need to deal with more indirect interventions, such as the CIA air assets in country (A-26s* & T-28s). I could envision a detachment of MiG-15UTIs at Stanleyville, ostensibly to train PRCongo pilots, "absolutely not" getting any Soviet instructors involved! On the ground, I'll leave it to those better versed.
Another possibility is that in the negotiations in the aftermath of the Cuban Missile Crisis, while chits like Jupiters are being traded, a wider dialogue is entered into by the superpowers. Perhaps the fate of various strategic backwaters is settled, with a moratorium on any US intervention in the Congo being exchanged for concessions elsewhere? Mobutu never gets throned in the first place - or has a change of spots!
* I know it should be B-26 by this stage but I personally adopt a first-come first-serve policy, so a B-26 is a Marauder until the end of time!