Check original Play! Framework migration documentation first:
a) change Play! version from
    <properties>
        ...
        <play2.version>2.5.18</play2.version>
    </properties>
to
    <properties>
        ...
        <play2.version>2.6.15</play2.version>
    </properties>
b) add dependency (unless compile-time or other dependency injection framework used)
        <dependency>
            <groupId>com.typesafe.play</groupId>
            <artifactId>play-guice_2.12</artifactId>
            <version>${play2.version}</version>
        </dependency>
c) optionally change HTTP server from play-netty-server
        <dependency>
            <groupId>com.typesafe.play</groupId>
            <artifactId>play-netty-server_2.12</artifactId>
            <version>${play2.version}</version>
            <scope>runtime</scope>
        </dependency>
to play-akka-http-server
        <dependency>
            <groupId>com.typesafe.play</groupId>
            <artifactId>play-akka-http-server_2.12</artifactId>
            <version>${play2.version}</version>
            <scope>runtime</scope>
        </dependency>
Previously only play-netty-server was available. Now play-akka-http-server is available and is used by default in SBT.
d) if using Java forms, add dependency (previously it was part of Play! Java module)
        <dependency>
            <groupId>com.typesafe.play</groupId>
            <artifactId>play-java-forms_2.12</artifactId>
            <version>${play2.version}</version>
        </dependency>
and optionally configure additional template imports
            <plugin>
                <groupId>com.google.code.play2-maven-plugin</groupId>
                <artifactId>play2-maven-plugin</artifactId>
                <version>${play2.plugin.version}</version>
                <extensions>true</extensions>
                <configuration>
                    ...
                    <templateAdditionalImports>play.data._ play.core.j.PlayFormsMagicForJava._</templateAdditionalImports>
                </configuration>
            </plugin>
e) if using Java JPA, upgrade Hibernate version to 5.x (e.g. 5.2.11.Final)
f) if using Ebean, upgrade play-ebean dependency version to 4.1.3
This will require some changes in model classes, e.g.: