Is there anyway to make a 70mm rocket with a speed around 1800-2000 m/s?
What type of rocket are you looking to produce? What delivery platform, or and what purpose are you looking to employ? These will determine the construction constraints for the product, as 70mm is a tight fit for most propulsion systems that would achieve this sort of velocity. Your target velocity is Mach 6 @ ISA AMSL, which is quite a fair clip.
Looking at missiles, the Stinger is 70mm wide with a 15-odd kg launch weight and manages up to Mach 2+, against the force of gravity. For a Hydra70, and at half the weight, 600m/s is more in the ballpark. The contract originally awarded to Lockheed and BEI Defence for ARS intended to replace Hydra originally called for a 1000m/s rocket of roughly the same dimensions. This was shelved in the 90's.
To achieve Mach 6 @ ISA (your target velocity) from a rocket that small to be launched from an existing Hydra-style launcher aboard an aircraft, or a stand-off type weapon, you would need to develop a rocket capable of accelerating in one single stage to Mach 6. This means you limit yourself to a minimum engagement range to allow enough burn time to acheive maximum velocity. However, on the flip side, this would likely take up most of your fuel load, and thus only has a short period to maintain this velocity. This means you shorten your engagement envelope on both sides making it a necessity to pick your ranges carefully. Next, you would spend a lot of effort developing high strength materials that weigh next to nothing to allow the performance envelope to increase to your target velocity. Next, you would need to ensure that all parts of the system can withstand the acceleration force AND the high airflow speed over the skin and control/stabilisation surfaces, costing even more money (requiring exotic materials). Then you would need to develop a fuel and oxygen source that would contain enough chemical energy to be released in a controlled fashion to provide the force necessary. Guidance would be difficult to fit in, as you are adding significant weight, and payload likewise would also be drastically limited. Afterwards, you have a very high chemical energy rocket motor that would need to be handled with extreme care lest it be inadvertently ignited while being handled/transported/fitted. This again limits the fuel, materials and other parts of the system.
So your drawbacks are:
-Cost and cost effectiveness of unit components;
-Limited payload;
-Limited guidance (if any);
-Potentially hazardous materials;
-Manufacturing process is likely to be very expensive (and possibly dangerous);
-Quality control would need to be infinitely strict;
-Limited engagement envelope.
For a unit launched from a land vehicle or a long range bomber, where length may not be limited so much, you would be using ramjet/scramjets to acheive sustained flight at that speed. You could potentially make a two-stage rocket with a booster phase to launch and provide initial acceleration to Mach 3+, which is discarded and a ramjet/scramjet takes over which would then accelerate to something similar to your target velocity for a 'cruise/terminal' phase. The main point of interest is whether your scramjet/ramjet would scale down to a size suitable for the application and still provide enough power. (I have seen very small ramjets, but whether they could go to Mach 6 is another matter)
The new set of major obstacles are (in addition to those mentioned previously):
-Cost per unit and consequently cost-effectiveness on a large scale (as you need to use exotic materials to construct scramjets/ramjets);
-Also dimensions as you would need quite a large reserve of fuel to produce a sustained flight of that length, so weight and length increase (to a point of diminishing returns, I suspect);
-Volatility of the materials used (as you'd be picking some pretty fancy chemicals to build, maintain and fuel your two-stage rocket);
-Manufacturing process (moreso than the previous model) could be dangerous or expensive or both;
-Some of the fuels used in odd types of combustion chambers could be toxic, corrosive or otherwise dangerous apart from their explosive nature, as you may have to invent your own fuels and oxygen source;
-Scalability of the second stage itself.
Generally, I see 70mm rockets as a cheap way to lay down stuff that goes 'bang', and a high velocity version doesn't fit with that rationale. However, if the system you are looking to deploy was a system that was simply an issue of requiring a fast-moving system but it was 70mm due to a space constriction of some kind in the launcher, then maybe it could be done if money were no object.
As I stated at the top: What type of rocket are you looking to produce? What delivery platform, or and what purpose are you looking to employ?