The internal security bodies of the civil war were reorganised and renamed a bewildering number of times. To add confusion, the Cheka operated outside any law or responsibility anyway, so looking at what its theoretical relationship was is no guide to how they operated in practice. I shall stick to discussing only units that might appear on an order of battle.
All these bodies were officially part of the NKVD, which was the over-riding Ministry for the Interior (not just a security organisation).
The Cheka was a purely internal security agency of early Soviet Russia. Though part of the NKVD, its troops were in no way similar to the NKVD forces of later times. It had plenty to do just dealing with counter-revolution in the period of the Civil War.
One of its roles was protecting the top Party men in Moscow. For this it had a small unit, including armoured cars. On 23 November 1919, with two companies and a cavalry squadron, it became the special battalion of the Cheka. These men really were the elite. Possibly they wore black leather.
But out in the provinces it just required guards, escorts and the heavier end of police functions (riot control, etc), which didn't require particularly good soldiers. Mostly it could borrow them, when required. The Bolsheviks had no intention of taking good troops away from the fronts fighting the Whites to perform internal security functions, and Trotsky was opposed to the Cheka having substantial armed forces in the first place. As a result, the bulk of the Cheka armed forces were ordinary troops leant to them, and were very much similar to ordinary Red Army, not specially trained or selected.
There were times when higher quality men were required. In particular putting down peasant revolts tended to be difficult with normal Red Army because the soldiers themselves tended to sympathise with the peasants. In these cases the Cheka had units that were unlikely to be affected – foreigners, and especially ones who didn't speak much Russian such as the Chinese, were particularly useful. The ChON could also be called on when a larger unit of reliable men was required on the internal fronts.
Most wargames rules give an elite "Cheka" ranking that doesn't correspond very well with the reality, especially in the front lines facing the White armies. Few Cheka units were seen on the external fronts, and they generally weren't high calibre anyway.
Use ChON, Kursanty, Internationalists or train landing parties to fill that elite role.
The Cheka itself had no uniforms, even in the 1919 regulations, because they were not part of the military. CheKa uniforms, as such, came in 1922.
In those regulations, not that the uniforms were common until many years later, internal security troops had normal Red Army uniforms with dark red distinctions and border troops had green (as their Tsarist equivalents had also had).
This was formed by combining various units tasked with guarding roads, rail lines etc.
Units of these appeared from time to time in the front lines, including an entire division on the Polish front in 1920. They were probably worse than most Red Army, since they lacked combat experience.
For example, a Cheka report for the problems suppressing anti-Bolshevik elements in the Ukraine in April 1920 says "The battalions of the internal troops are badly equipped and leave much to be desired in all ways". The eventual suppression of Makhno was largely done by regular cavalry divisions.
The Directives of the Red Army, when counting its armed forces, described the internal security units as "Troops in the combat composition of the armed forces for auxiliary purposes, such as: guard units, troops of the Cheka, railway security, food armies, etc., as troops of no combat significance, the number of which was 105,500 bayonets and sabres."
This body briefly existed late in the civil war period, combined the various internal security organs: CheKa, VOKhR, border guards etc. As before, it was not in the job of training troops, and its units were far from elite.
Landis says (p 154), "While some regular Red Army troops were brought to the province, many of the forces that were assigned to Tambov in early 1921 were former VNUS units, which were frequently filled with "second-chance men" who had previously deserted and who were undertrained, poorly supplied, and unaccustomed to counterinsurgency operations".
Very occasionally zagraditelnye otriady or "blocking detachments" were used in the Civil War period. Their job was to follow front line units, with the task of arresting or shooting anyone withdrawing from the battle without permission. Most occasions I have seen them has been associated with KomandArm Tukhachevski, though I've also seen a couple listed for the campaign against Iudenich. Regardless, they were uncommon at this time – the extremely low troop densities and fast movements of the front made such units only useful to hold key points.
Cheka troops were sometimes used for this purpose, but often it was normal Red Army.
Blocking units were also used to enforce the Bolsheviks' food policy, which made them wildly unpopular.
Probably the most hated of all Soviet units. These were the units sent into the countryside to force the peasants at gun point to hand over their "surplus". Made up to city dwellers, one suspects, as peasants simply would not do it.
Of very little military value, they were occasionally seen in emergencies at the front line.
A useful quick reference for fighting through the various acronyms and rapid changes in organisation of the internal security apparatus in the early years of Soviet power.
E.A. Kobeleva. The Military Functions of the State Security Organs of the Urals during the Civil War , Perm State Pedagogical University
A short article on the actions of the Perm area Cheka's military forces. While several units were sent to the front, it is noticeable how small they were and how the first appears to have disintegrated in combat. The original Russian is available at the Perm Historical Archive.
Erik C. Landis. Bandits and Partisans: the Antonov Movement in the Russian Civil War. 2008, University of Pittsburg Press.
A fascinating book about an interesting episode. He relates how poorly the internal security troops performed against Antonov, and indeed how the VOKhR/VnuS troops were stated to be primarily intended for non-combat roles. They deserted at a very high rate, which hardly suggests high morale.
Despite being the centre of a massive rebellion, there is almost no mention of Cheka troops, and it was largely regular cavalry and motorised forces that took the active combat roles in the eventual suppression.
The Case of the Surrender of Tambov from the State Archive of Social and Political History of the Tambov Region.
This relates how a garrison of kursanty, ChON, Cheka and VOKhR (plus regular Red Army, including armour cars) retreated rather than face Mamontov's Cossacks.
From "The Yekaterinburg GubChEKa in 1919–1922" by Sergei Konstantinov, Ural State Law University
The Cheka needed armed formations attached. Thus, on 21 August  a memorandum was submitted to the Provincial RMS proposing the formation and financing of internal security troops with all the weapons and specialties required for a combat battalion. The number of troops was planned to be 600, with 450 bayonets and the remainder being command staff, liaison officers, scouts, machine gunners and doctors. It was proposed to station 200 men in Yekaterinburg, while the rest – for the purpose of rapid redeployment – were to be dispersed to large industrial areas.