• We cannot Create or Update Custom Metadata records using DML statement.
• We can Create or Update Custom Metadata records using apex Metadata deployment.
• To do the Metadata deployment using apex, it requires a callback class with Metadata.DeployCallback interface.

Sample Metadata Deployment Callback Class:

public class CustomMetadataCallBack implements Metadata.DeployCallback {
    
    //Inteface method 
    public void handleResult(Metadata.DeployResult result, Metadata.DeployCallbackContext context) {
        if (result.status == Metadata.DeployStatus.Succeeded) {
            //Success
        } else {
            //Failed
        }
    }
}

Here I have created a Custom Metadata Type (Tax Setting), to save Tax related information for an Application.

Here I have created a class with Metadata.DeployCallback interface, and two methods to create and update Custom Metadata Type records.

Apex Class:

public class CustomMetadataUtils implements Metadata.DeployCallback {
    
    //Inteface method 
    public void handleResult(Metadata.DeployResult result, Metadata.DeployCallbackContext context) {
        if (result.status == Metadata.DeployStatus.Succeeded) {
            //Success
            System.debug('Success Result-' + result);
        } else {
            //Failed
            System.debug('Failed Result-' + result);
        }
    }
    
    //Create Custom Metadata record
    public static void createCustomMetadata(String metdataName, String label, Map<String, Object> metadataFieldValueMap){
        String recordDevName = label.replaceAll(' ', '_');
        Metadata.CustomMetadata cMetadata = new Metadata.CustomMetadata();
        cMetadata.fullName = metdataName + '.' + recordDevName;
        cMetadata.label = label;
        
        for(String key : metadataFieldValueMap.keySet()){
            Metadata.CustomMetadataValue cMetadataValue = new Metadata.CustomMetadataValue();
            cMetadataValue.Field = key;
            cMetadataValue.Value = metadataFieldValueMap.get(key); 
            cMetadata.values.add(cMetadataValue);
        }
        
        Metadata.DeployContainer mdContainer = new Metadata.DeployContainer();
        mdContainer.addMetadata(cMetadata);
        CustomMetadataUtils callback = new CustomMetadataUtils();
        Id jobId = Metadata.Operations.enqueueDeployment(mdContainer, callback);
    }
    
    //Update Custom Metadata record
    public static void updateCustomMetadata(String metdataName, String recordDevName, String label, Map<String, Object> metadataFieldValueMap){
        Metadata.CustomMetadata cMetadata = new Metadata.CustomMetadata();
        cMetadata.fullName = metdataName + '.' + recordDevName;
        cMetadata.label = label;
        
        for(String key : metadataFieldValueMap.keySet()){
            Metadata.CustomMetadataValue cMetadataValue = new Metadata.CustomMetadataValue();
            cMetadataValue.Field = key;
            cMetadataValue.Value = metadataFieldValueMap.get(key); 
            cMetadata.values.add(cMetadataValue);
        }
        
        Metadata.DeployContainer mdContainer = new Metadata.DeployContainer();
        mdContainer.addMetadata(cMetadata);
        CustomMetadataUtils callback = new CustomMetadataUtils();
        Id jobId = Metadata.Operations.enqueueDeployment(mdContainer, callback);
    }
}

To test above class methods, you can run below code in developer console.
Create Custom Metadata Record:

Map<String, Object> metadataFieldValueMap = new Map<String, Object>();
metadataFieldValueMap.put('TaxPercent__c', 4);
CustomMetadataUtils.createCustomMetadata('SalesTaxSetting__mdt', 'Value Added Tax', metadataFieldValueMap);

Update Custom Metadata Record:

Map<String, Object> metadataFieldValueMap = new Map<String, Object>();
metadataFieldValueMap.put('TaxPercent__c', 15);
CustomMetadataUtils.updateCustomMetadata('SalesTaxSetting__mdt','Value_Added_Tax', 'Value Added Tax',metadataFieldValueMap);

After deployed the Custom Metadata, you can check the deployment status.
Go to Setup | Deploy | Deployment Status | Check the deployment status