In order to try and resolve the dispute without having to pursue court action, a formal letter before action should be drafted and sent, to demand payment within 14 days and say that if they do not pay you, you will issue county court proceedings against them.
See attached template as an example of what to say. Please let me know if you cannot see the letter which I have uploaded. You will need to tailor this letter to your situation. The pre action protocol confirms you should send the letter to give the other party a chance to avoid court action and to pay you. The courts encourage compliance with the protocol as it can result in a resolution without having to involve the court.
If you decided to issue a claim, you would then need to register at https://www.moneyclaim.gov.uk/web/mcol/welcome so that you are ready to issue the claim in the event they dispute the claim and do not pay you. The website is very user-friendly and you would not need a lawyer to use the money claim site.
Claims with a value of under £10,000 are classed as a "small claim", so legal costs are generally not recoverable and the matter may be dealt with on paper by a Judge, not at a hearing. This means the parties are on an equal footing, so you don’t need to worry about legal costs if you lost.
A hearing may be necessary if the court thinks that oral evidence is required to dispose of the case.
Here is a user guide for the money claim online site: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/1020660/mcol-userguide-eng.pdf
The court will then issue the claim and they will send you "notice of issue". The papers are served on the defendant who then has 14 days to acknowledge the claim - they do this by filling out an acknowledgment of service and they post it to the court. They indicate their intention when they do this, i.e. whether they admit the claim in full or partly, or if they deny the claim. If they want to defend the claim then their defence is due by 28 days from the date the court served them with the papers. The central court processing centre then sends the claim to the defendant's home county court for case management and directions - the directions will give a list of dates which you both must comply with.
If there is no settlement then the claim will be dealt with at a final hearing which takes anything from 9 to 12 months from when you start the claim - longer if the claim is higher value. You can pursue the claim yourself or use a law firm. For the hearing you can use an advocate if you wish, though it's not compulsory. I have details of law firms and advocacy agencies if you would like those. Though in a small claim you won't be able to recover their charges from your opponent. A small claims hearing is easy to do, it's quite informal and no lawyer is required.
You would claim the sum for the loss, the court issue fee (details of the money claim fees are listed here at page 5: EX50 - Civil and Family Court fees) and court interest which is 8% calculated on a daily rate from the date of loss to date of court judgment.
So for example if you were owed £10,000, interest would be £2.20 per day, which you can also claim.
The site allows you to calculate the interest and add it to the claim. If a hearing is required then there is a fee for this too - see page 7 of the previous link for details. Again, that fee is recoverable if you win.
If you win then once you have the CCJ from the court the defendant has 14 days to pay in full. If they do not then it gets registered with the credit agencies after 30 days. You can also enforce the CCJ with the county court bailiffs or transfer the debt to the High Court for a small additional fee assuming the total amount owed is at least £600 and you can use the high court enforcement officers who have greater powers than county court bailiffs. The transfer fee is added on to the debt and payable by the defendant.
There are other enforcement methods which I can help with, including bank account freeze, charging order on their property (and then apply to force a sale), apply to summons them to court for questioning, attachment of earnings order against their employer (if employed), apply to bankrupt them if they are an individual or sole trader and they owe £5,000 or more - all of which can ensure you are actually repaid the money.
You may find they just pay you after receiving the letter before action – hopefully they will want to avoid litigation.
As for your likelihood of winning, in a civil claim you only need to show there was a likely breach of contract - the burden of proof is only 51% or "more likely than not".